matrix element - digital

arXiv:0811.1062v2 [hep-ex] 15 Apr 2009
Top quark mass measurement in the tt̄ all hadronic
√ channel using a matrix element
technique in pp̄ collisions at s = 1.96 TeV
T. Aaltonen,24 J. Adelman,14 T. Akimoto,56 B. Álvarez González,12 S. Ameriow ,44 D. Amidei,35 A. Anastassov,39
A. Annovi,20 J. Antos,15 G. Apollinari,18 A. Apresyan,49 T. Arisawa,58 A. Artikov,16 W. Ashmanskas,18 A. Attal,4
A. Aurisano,54 F. Azfar,43 P. Azzurriz ,47 W. Badgett,18 A. Barbaro-Galtieri,29 V.E. Barnes,49 B.A. Barnett,26
V. Bartsch,31 G. Bauer,33 P.-H. Beauchemin,34 F. Bedeschi,47 D. Beecher,31 S. Behari,26 G. Bellettinix ,47
J. Bellinger,60 D. Benjamin,17 A. Beretvas,18 J. Beringer,29 A. Bhatti,51 M. Binkley,18 D. Bisellow ,44 I. Bizjakcc ,31
R.E. Blair,2 C. Blocker,7 B. Blumenfeld,26 A. Bocci,17 A. Bodek,50 V. Boisvert,50 G. Bolla,49 D. Bortoletto,49
J. Boudreau,48 A. Boveia,11 B. Braua ,11 A. Bridgeman,25 L. Brigliadori,44 C. Bromberg,36 E. Brubaker,14
J. Budagov,16 H.S. Budd,50 S. Budd,25 S. Burke,18 K. Burkett,18 G. Busettow ,44 P. Busseyk ,22 A. Buzatu,34
K. L. Byrum,2 S. Cabrerau ,17 C. Calancha,32 M. Campanelli,36 M. Campbell,35 F. Canelli14 ,18 A. Canepa,46
B. Carls,25 D. Carlsmith,60 R. Carosi,47 S. Carrillom,19 S. Carron,34 B. Casal,12 M. Casarsa,18 A. Castrov ,6
P. Catastiniy ,47 D. Cauzbb ,55 V. Cavalierey ,47 M. Cavalli-Sforza,4 A. Cerri,29 L. Cerriton ,31 S.H. Chang,28
Y.C. Chen,1 M. Chertok,8 G. Chiarelli,47 G. Chlachidze,18 F. Chlebana,18 K. Cho,28 D. Chokheli,16
J.P. Chou,23 G. Choudalakis,33 S.H. Chuang,53 K. Chung,13 W.H. Chung,60 Y.S. Chung,50 T. Chwalek,27
C.I. Ciobanu,45 M.A. Ciocciy ,47 A. Clark,21 D. Clark,7 G. Compostella,44 M.E. Convery,18 J. Conway,8
M. Cordelli,20 G. Cortianaw ,44 C.A. Cox,8 D.J. Cox,8 F. Cresciolix,47 C. Cuenca Almenaru ,8 J. Cuevasr ,12
R. Culbertson,18 J.C. Cully,35 D. Dagenhart,18 M. Datta,18 T. Davies,22 P. de Barbaro,50 S. De Cecco,52
A. Deisher,29 G. De Lorenzo,4 M. Dell’Orsox,47 C. Deluca,4 L. Demortier,51 J. Deng,17 M. Deninno,6
P.F. Derwent,18 G.P. di Giovanni,45 C. Dionisiaa ,52 B. Di Ruzzabb ,55 J.R. Dittmann,5 M. D’Onofrio,4 S. Donatix ,47
P. Dong,9 J. Donini,44 T. Dorigo,44 S. Dube,53 J. Efron,40 A. Elagin,54 R. Erbacher,8 D. Errede,25 S. Errede,25
R. Eusebi,18 H.C. Fang,29 S. Farrington,43 W.T. Fedorko,14 R.G. Feild,61 M. Feindt,27 J.P. Fernandez,32
C. Ferrazzaz ,47 R. Field,19 G. Flanagan,49 R. Forrest,8 M.J. Frank,5 M. Franklin,23 J.C. Freeman,18 I. Furic,19
M. Gallinaro,52 J. Galyardt,13 F. Garberson,11 J.E. Garcia,21 A.F. Garfinkel,49 K. Genser,18 H. Gerberich,25
D. Gerdes,35 A. Gessler,27 S. Giaguaa ,52 V. Giakoumopoulou,3 P. Giannetti,47 K. Gibson,48 J.L. Gimmell,50
C.M. Ginsburg,18 N. Giokaris,3 M. Giordanibb ,55 P. Giromini,20 M. Giuntax ,47 G. Giurgiu,26 V. Glagolev,16
D. Glenzinski,18 M. Gold,38 N. Goldschmidt,19 A. Golossanov,18 G. Gomez,12 G. Gomez-Ceballos,33
M. Goncharov,33 O. González,32 I. Gorelov,38 A.T. Goshaw,17 K. Goulianos,51 A. Greselew ,44 S. Grinstein,23
C. Grosso-Pilcher,14 R.C. Group,18 U. Grundler,25 J. Guimaraes da Costa,23 Z. Gunay-Unalan,36 C. Haber,29
K. Hahn,33 S.R. Hahn,18 E. Halkiadakis,53 B.-Y. Han,50 J.Y. Han,50 F. Happacher,20 K. Hara,56 D. Hare,53
M. Hare,57 S. Harper,43 R.F. Harr,59 R.M. Harris,18 M. Hartz,48 K. Hatakeyama,51 C. Hays,43 M. Heck,27
A. Heijboer,46 J. Heinrich,46 C. Henderson,33 M. Herndon,60 J. Heuser,27 S. Hewamanage,5 D. Hidas,17
C.S. Hillc ,11 D. Hirschbuehl,27 A. Hocker,18 S. Hou,1 M. Houlden,30 S.-C. Hsu,29 B.T. Huffman,43 R.E. Hughes,40
U. Husemann,36 M. Hussein,36 U. Husemann,61 J. Huston,36 J. Incandela,11 G. Introzzi,47 M. Ioriaa ,52 A. Ivanov,8
E. James,18 B. Jayatilaka,17 E.J. Jeon,28 M.K. Jha,6 S. Jindariani,18 W. Johnson,8 M. Jones,49 K.K. Joo,28
S.Y. Jun,13 J.E. Jung,28 T.R. Junk,18 T. Kamon,54 D. Kar,19 P.E. Karchin,59 Y. Kato,42 R. Kephart,18 J. Keung,46
V. Khotilovich,54 B. Kilminster,18 D.H. Kim,28 H.S. Kim,28 H.W. Kim,28 J.E. Kim,28 M.J. Kim,20 S.B. Kim,28
S.H. Kim,56 Y.K. Kim,14 N. Kimura,56 L. Kirsch,7 S. Klimenko,19 B. Knuteson,33 B.R. Ko,17 K. Kondo,58
D.J. Kong,28 J. Konigsberg,19 A. Korytov,19 A.V. Kotwal,17 M. Kreps,27 J. Kroll,46 D. Krop,14 N. Krumnack,5
M. Kruse,17 V. Krutelyov,11 T. Kubo,56 T. Kuhr,27 N.P. Kulkarni,59 M. Kurata,56 S. Kwang,14 A.T. Laasanen,49
S. Lami,47 S. Lammel,18 M. Lancaster,31 R.L. Lander,8 K. Lannonq ,40 A. Lath,53 G. Latinoy ,47 I. Lazzizzeraw ,44
T. LeCompte,2 E. Lee,54 H.S. Lee,14 S.W. Leet ,54 S. Leone,47 J.D. Lewis,18 C.-S. Lin,29 J. Linacre,43 M. Lindgren,18
E. Lipeles,46 A. Lister,8 D.O. Litvintsev,18 C. Liu,48 T. Liu,18 N.S. Lockyer,46 A. Loginov,61 M. Loretiw ,44
L. Lovas,15 D. Lucchesiw ,44 C. Luciaa ,52 J. Lueck,27 P. Lujan,29 P. Lukens,18 G. Lungu,51 L. Lyons,43 J. Lys,29
R. Lysak,15 D. MacQueen,34 R. Madrak,18 K. Maeshima,18 K. Makhoul,33 T. Maki,24 P. Maksimovic,26 S. Malde,43
S. Malik,31 G. Mancae ,30 A. Manousakis-Katsikakis,3 F. Margaroli,49 C. Marino,27 C.P. Marino,25 A. Martin,61
V. Martinl ,22 M. Martı́nez,4 R. Martı́nez-Balları́n,32 T. Maruyama,56 P. Mastrandrea,52 T. Masubuchi,56
M. Mathis,26 M.E. Mattson,59 P. Mazzanti,6 K.S. McFarland,50 P. McIntyre,54 R. McNultyj ,30 A. Mehta,30
P. Mehtala,24 A. Menzione,47 P. Merkel,49 C. Mesropian,51 T. Miao,18 N. Miladinovic,7 R. Miller,36 C. Mills,23
M. Milnik,27 A. Mitra,1 G. Mitselmakher,19 H. Miyake,56 N. Moggi,6 C.S. Moon,28 R. Moore,18 M.J. Morellox ,47
J. Morlok,27 P. Movilla Fernandez,18 J. Mülmenstädt,29 A. Mukherjee,18 Th. Muller,27 R. Mumford,26 P. Murat,18
M. Mussiniv ,6 J. Nachtman,18 Y. Nagai,56 A. Nagano,56 J. Naganoma,56 K. Nakamura,56 I. Nakano,41 A. Napier,57
2
V. Necula,17 J. Nett,60 C. Neuv ,46 M.S. Neubauer,25 S. Neubauer,27 J. Nielseng ,29 L. Nodulman,2
M. Norman,10 O. Norniella,25 E. Nurse,31 L. Oakes,43 S.H. Oh,17 Y.D. Oh,28 I. Oksuzian,19 T. Okusawa,42
R. Orava,24 S. Pagan Grisow ,44 E. Palencia,18 V. Papadimitriou,18 A. Papaikonomou,27 A.A. Paramonov,14
B. Parks,40 S. Pashapour,34 J. Patrick,18 G. Paulettabb ,55 M. Paulini,13 C. Paus,33 T. Peiffer,27 D.E. Pellett,8
A. Penzo,55 T.J. Phillips,17 G. Piacentino,47 E. Pianori,46 L. Pinera,19 K. Pitts,25 C. Plager,9 L. Pondrom,60
O. Poukhov∗,16 N. Pounder,43 F. Prakoshyn,16 A. Pronko,18 J. Proudfoot,2 F. Ptohosi ,18 E. Pueschel,13
G. Punzix ,47 J. Pursley,60 J. Rademackerc,43 A. Rahaman,48 V. Ramakrishnan,60 N. Ranjan,49 I. Redondo,32
P. Renton,43 M. Renz,27 M. Rescigno,52 S. Richter,27 F. Rimondiv ,6 L. Ristori,47 A. Robson,22 T. Rodrigo,12
T. Rodriguez,46 E. Rogers,25 S. Rolli,57 R. Roser,18 M. Rossi,55 R. Rossin,11 P. Roy,34 A. Ruiz,12 J. Russ,13
V. Rusu,18 A. Safonov,54 W.K. Sakumoto,50 O. Saltó,4 L. Santibb ,55 S. Sarkaraa,52 L. Sartori,47 K. Sato,18
A. Savoy-Navarro,45 P. Schlabach,18 A. Schmidt,27 E.E. Schmidt,18 M.A. Schmidt,14 M.P. Schmidt∗ ,61
M. Schmitt,39 T. Schwarz,8 L. Scodellaro,12 A. Scribanoy ,47 F. Scuri,47 A. Sedov,49 S. Seidel,38 Y. Seiya,42
A. Semenov,16 L. Sexton-Kennedy,18 F. Sforza,47 A. Sfyrla,25 S.Z. Shalhout,59 T. Shears,30 P.F. Shepard,48
M. Shimojimap ,56 S. Shiraishi,14 M. Shochet,14 Y. Shon,60 I. Shreyber,37 A. Sidoti,47 P. Sinervo,34 A. Sisakyan,16
A.J. Slaughter,18 J. Slaunwhite,40 K. Sliwa,57 J.R. Smith,8 F.D. Snider,18 R. Snihur,34 A. Soha,8 S. Somalwar,53
V. Sorin,36 J. Spalding,18 T. Spreitzer,34 P. Squillaciotiy ,47 M. Stanitzki,61 R. St. Denis,22 B. Stelzer,34
O. Stelzer-Chilton,34 D. Stentz,39 J. Strologas,38 G.L. Strycker,35 D. Stuart,11 J.S. Suh,28 A. Sukhanov,19
I. Suslov,16 T. Suzuki,56 A. Taffardf ,25 R. Takashima,41 Y. Takeuchi,56 R. Tanaka,41 M. Tecchio,35 P.K. Teng,1
K. Terashi,51 J. Thomh ,18 A.S. Thompson,22 G.A. Thompson,25 E. Thomson,46 P. Tipton,61 P. Ttito-Guzmán,32
S. Tkaczyk,18 D. Toback,54 S. Tokar,15 K. Tollefson,36 T. Tomura,56 D. Tonelli,18 S. Torre,20 D. Torretta,18
P. Totarobb ,55 S. Tourneur,45 M. Trovato,47 S.-Y. Tsai,1 Y. Tu,46 N. Turiniy ,47 F. Ukegawa,56 S. Vallecorsa,21
N. van Remortelb ,24 A. Varganov,35 E. Vatagaz ,47 F. Vázquezm ,19 G. Velev,18 C. Vellidis,3 M. Vidal,32 R. Vidal,18
I. Vila,12 R. Vilar,12 T. Vine,31 M. Vogel,38 I. Volobouevt ,29 G. Volpix ,47 P. Wagner,46 R.G. Wagner,2
R.L. Wagner,18 W. Wagner,27 J. Wagner-Kuhr,27 T. Wakisaka,42 R. Wallny,9 S.M. Wang,1 A. Warburton,34
D. Waters,31 M. Weinberger,54 J. Weinelt,27 W.C. Wester III,18 B. Whitehouse,57 D. Whitesonf ,46 A.B. Wicklund,2
E. Wicklund,18 S. Wilbur,14 G. Williams,34 H.H. Williams,46 P. Wilson,18 B.L. Winer,40 P. Wittichh ,18
S. Wolbers,18 C. Wolfe,14 T. Wright,35 X. Wu,21 F. Würthwein,10 S. Xie,33 A. Yagil,10 K. Yamamoto,42
J. Yamaoka,17 U.K. Yango ,14 Y.C. Yang,28 W.M. Yao,29 G.P. Yeh,18 J. Yoh,18 K. Yorita,58 T. Yoshida,42
G.B. Yu,50 I. Yu,28 S.S. Yu,18 J.C. Yun,18 L. Zanelloaa ,52 A. Zanetti,55 X. Zhang,25 Y. Zhengd ,9 and S. Zucchelliv ,6
(CDF Collaboration†)
1
Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, Republic of China
2
Argonne National Laboratory, Argonne, Illinois 60439
3
University of Athens, 157 71 Athens, Greece
4
Institut de Fisica d’Altes Energies, Universitat Autonoma de Barcelona, E-08193, Bellaterra (Barcelona), Spain
5
Baylor University, Waco, Texas 76798
6
Istituto Nazionale di Fisica Nucleare Bologna, v University of Bologna, I-40127 Bologna, Italy
7
Brandeis University, Waltham, Massachusetts 02254
8
University of California, Davis, Davis, California 95616
9
University of California, Los Angeles, Los Angeles, California 90024
10
University of California, San Diego, La Jolla, California 92093
11
University of California, Santa Barbara, Santa Barbara, California 93106
12
Instituto de Fisica de Cantabria, CSIC-University of Cantabria, 39005 Santander, Spain
13
Carnegie Mellon University, Pittsburgh, PA 15213
14
Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637
15
Comenius University, 842 48 Bratislava, Slovakia; Institute of Experimental Physics, 040 01 Kosice, Slovakia
16
Joint Institute for Nuclear Research, RU-141980 Dubna, Russia
17
Duke University, Durham, North Carolina 27708
18
Fermi National Accelerator Laboratory, Batavia, Illinois 60510
19
University of Florida, Gainesville, Florida 32611
20
Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, I-00044 Frascati, Italy
21
University of Geneva, CH-1211 Geneva 4, Switzerland
22
Glasgow University, Glasgow G12 8QQ, United Kingdom
23
Harvard University, Cambridge, Massachusetts 02138
24
Division of High Energy Physics, Department of Physics,
University of Helsinki and Helsinki Institute of Physics, FIN-00014, Helsinki, Finland
25
University of Illinois, Urbana, Illinois 61801
26
The Johns Hopkins University, Baltimore, Maryland 21218
27
Institut für Experimentelle Kernphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany
3
28
Center for High Energy Physics: Kyungpook National University,
Daegu 702-701, Korea; Seoul National University, Seoul 151-742,
Korea; Sungkyunkwan University, Suwon 440-746,
Korea; Korea Institute of Science and Technology Information, Daejeon,
305-806, Korea; Chonnam National University, Gwangju, 500-757, Korea
29
Ernest Orlando Lawrence Berkeley National Laboratory, Berkeley, California 94720
30
University of Liverpool, Liverpool L69 7ZE, United Kingdom
31
University College London, London WC1E 6BT, United Kingdom
32
Centro de Investigaciones Energeticas Medioambientales y Tecnologicas, E-28040 Madrid, Spain
33
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
34
Institute of Particle Physics: McGill University, Montréal, Québec,
Canada H3A 2T8; Simon Fraser University, Burnaby, British Columbia,
Canada V5A 1S6; University of Toronto, Toronto, Ontario,
Canada M5S 1A7; and TRIUMF, Vancouver, British Columbia, Canada V6T 2A3
35
University of Michigan, Ann Arbor, Michigan 48109
36
Michigan State University, East Lansing, Michigan 48824
37
Institution for Theoretical and Experimental Physics, ITEP, Moscow 117259, Russia
38
University of New Mexico, Albuquerque, New Mexico 87131
39
Northwestern University, Evanston, Illinois 60208
40
The Ohio State University, Columbus, Ohio 43210
41
Okayama University, Okayama 700-8530, Japan
42
Osaka City University, Osaka 588, Japan
43
University of Oxford, Oxford OX1 3RH, United Kingdom
44
Istituto Nazionale di Fisica Nucleare, Sezione di Padova-Trento, w University of Padova, I-35131 Padova, Italy
45
LPNHE, Universite Pierre et Marie Curie/IN2P3-CNRS, UMR7585, Paris, F-75252 France
46
University of Pennsylvania, Philadelphia, Pennsylvania 19104
47
Istituto Nazionale di Fisica Nucleare Pisa, x University of Pisa,
y
University of Siena and z Scuola Normale Superiore, I-56127 Pisa, Italy
48
University of Pittsburgh, Pittsburgh, Pennsylvania 15260
49
Purdue University, West Lafayette, Indiana 47907
50
University of Rochester, Rochester, New York 14627
51
The Rockefeller University, New York, New York 10021
52
Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1,
aa
Sapienza Università di Roma, I-00185 Roma, Italy
53
Rutgers University, Piscataway, New Jersey 08855
54
Texas A&M University, College Station, Texas 77843
55
Istituto Nazionale di Fisica Nucleare Trieste/Udine, bb University of Trieste/Udine, Italy
56
University of Tsukuba, Tsukuba, Ibaraki 305, Japan
57
Tufts University, Medford, Massachusetts 02155
58
Waseda University, Tokyo 169, Japan
59
Wayne State University, Detroit, Michigan 48201
60
University of Wisconsin, Madison, Wisconsin 53706
61
Yale University, New Haven, Connecticut 06520
(Dated: April 15, 2009)
We present a measurement of the√top quark mass in the all-hadronic channel (tt̄ → bb̄ q1 q¯2 q3 q¯4 )
using 943 pb−1 of pp̄ collisions at s = 1.96 TeV collected at the CDF II detector at Fermilab
(CDF). We apply the standard model production and decay matrix-element (ME) to tt candidate
events. We calculate per-event probability densities according to the ME calculation and construct
template models of signal and background. The scale of the jet energy is calibrated using additional
templates formed with the invariant mass of pairs of jets. These templates form an overall likelihood
function that depends on the top quark mass and on the jet energy scale (JES). We estimate both
by maximizing this function. Given 72 observed events, we measure a top quark mass of 171.1 ± 3.7
(stat.+JES) ± 2.1 (syst.) GeV/c2 . The combined uncertainty on the top quark mass is 4.3 GeV/c2 .
PACS numbers: 14.65.Ha, 12.15.Ff, 13.85.Ni
∗ Deceased
visitors from a University of Massachusetts Amherst,
Amherst, Massachusetts 01003, b Universiteit Antwerpen, B-2610
† With
Antwerp, Belgium, c University of Bristol, Bristol BS8 1TL,
United Kingdom, d Chinese Academy of Sciences, Beijing 100864,
China, e Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari,
4
I.
INTRODUCTION
The top quark plays an important role in particle
physics. Being the heaviest observed elementary particle results in large contributions to electroweak radiative
corrections and provides a constraint on the mass of the
elusive Higgs boson. More accurate measurements of the
top quark mass are important for precision tests of the
standard model. In addition, having a Yukawa coupling
close to unity may indicate a special role for this quark in
electroweak symmetry breaking. Increasing the precision
on the mass of the top quark is central not only for the
standard model, but also for other theoretical scenarios
beyond the standard model.
At the Tevatron the top quark is produced most frequently via the strong interaction yielding a top/anti-top
pair. Once produced, the top quark decays into a b quark
and a W boson about 99% of the time according to the
standard model. Based on the decay mode of the two
W bosons the tt̄ events can be divided in three channels: the dilepton channel when both W bosons decay
to leptons; the lepton+jets channel when one W boson
decays to leptons and the other one decays to hadrons;
and the all-hadronic channel when both W bosons decay
to hadrons.
This paper reports a measurement of the top quark
mass in the all-hadronic channel using 943 pb−1 collected
with the upgraded CDF II detector at Fermilab. In Section II we give a brief description of the CDF II detector.
The all-hadronic final state consists of six jets, two of
which are due to the hadronization of b quarks. The large
QCD background and jet-parton combinatorics make
measurements more difficult in this channel than in the
others. However, because there are no unobserved finalstate particles, it is possible to fully reconstruct allhadronic events. In order to enhance the tt̄ content over
the background, special selection criteria are imposed on
the kinematics and topology of the events. In Section III
we give more details on this selection.
Previous mass measurements of the top quark in the
09042 Monserrato (Cagliari), Italy, f University of California
Irvine, Irvine, CA 92697, g University of California Santa Cruz,
Santa Cruz, CA 95064, h Cornell University, Ithaca, NY 14853,
i University of Cyprus, Nicosia CY-1678, Cyprus, j University
College Dublin, Dublin 4, Ireland, k Royal Society of Edinburgh/Scottish Executive Support Research Fellow, l University of
Edinburgh, Edinburgh EH9 3JZ, United Kingdom, m Universidad
Iberoamericana, Mexico D.F., Mexico, n Queen Mary, University
of London, London, E1 4NS, England, o University of Manchester,
Manchester M13 9PL, England, p Nagasaki Institute of Applied Science, Nagasaki, Japan, q University of Notre Dame, Notre Dame,
IN 46556, r University de Oviedo, E-33007 Oviedo, Spain, s Texas
Tech University, Lubbock, TX 79409, t IFIC(CSIC-Universitat de
Valencia), 46071 Valencia, Spain, u University of Virginia, Charlottesville, VA 22904, cc On leave from J. Stefan Institute, Ljubljana, Slovenia,
all-hadronic channel were performed at CDF in both Run
I [1] and Run II [2]. For the first time in this channel,
we measure the mass of the top quark utilizing a technique that uses the matrix element for tt̄ production and
decay. The details of the matrix element calculation and
implementation are given in Section IV.
The matrix element is used to calculate a probability
for each candidate event to be produced via the standard model tt̄ production mechanism. In principle, the
mass of the top quark can be determined by maximizing
this probability, and such a technique was successfully
applied before at CDF in the lepton+jets channel [3] and
in the dilepton channel [4]. In this analysis we take a new
approach in that we calculate the matrix element probability in samples of simulated tt̄ events to build and to
parameterize top mass templates. These are distributions that depend on the mass of the top quark, unlike
the templates for background events whose modeling is
described in Section V. The measured value of the mass
of the top quark corresponds to a tt̄ template whose mixture with a background template best fits the data. In
Section VI we give more details on how these templates
are built.
Besides considering a matrix element for a different
tt̄ decay channel, in this analysis the matrix element is
computed differently than in the aforementioned analyses
in the leptonic channels. Also, the features of the matrix
element probability are exploited to improve the event
selection.
The uncertainty on the jet energy scale has the largest
contribution to the total uncertainty in most top quark
mass measurements. In order to minimize this effect,
we perform an in situ calibration of the jet energy scale
using the invariant mass of pairs of light flavor jets. For
tt̄ events this variable is correlated with the mass of the
W boson, and it is sensitive to variations in jet energy
scale. Using this invariant mass we build the dijet mass
templates, and we use them for the calibration of the jet
energy scale as shown in Section VI. This procedure,
used previously at CDF for the mass measurement of
the top quark in the lepton+jets channel [5], is used for
the first time in the all-hadronic channel in the analysis
described in this paper.
The result of the data fit is the topic of Section VII,
while in Section VIII the associated systematic uncertainties are described. Finally, Section IX concludes the
paper.
II.
DETECTOR
The CDF II detector is an azimuthally and forwardbackward symmetric apparatus designed to study pp̄ collisions at the Tevatron. It is a general purpose detector
which combines precision particle tracking with fast projective calorimetry and fine grained muon detection.
The CDF coordinate system is right handed, with z
axis tangent to the Tevatron ring and pointing in the di-
5
rection of the proton beam. The x and y coordinates of a
left-handed x,y, z Cartesian reference system are defined
pointing outward and upward from the Tevatron ring,
respectively. The azimuthal angle φ is measured relative
to the x axis in the transverse plane. The polar angle
θ is measured from the proton direction and is typically
expressed as pseudorapidity η = −ln(tan θ2 ). We define
transverse energy as ET = Esinθ and transverse momentum as pT = psinθ where E is the energy measured in
the calorimeter and p is the magnitude of the momentum
measured by the tracking system.
Tracking systems are contained in a superconducting
solenoid, 1.5 m in radius and 4.8 m in length, which generates a 1.4 T magnetic field parallel to the beam axis.
The calorimeter surrounds the solenoid. A more complete description of the CDF II detector can be found in
Ref. [6]. The main features of the detector systems are
summarized below.
The tracking system consists of a silicon microstrip system and an open-cell wire drift chamber that surrounds
the silicon. The silicon microstrip system consists of eight
layers in a cylindrical geometry that extends from a radius of r = 1.35 cm from the beam line to r = 29 cm. The
layer closest to the beam pipe is a radiation-hard, single
sided detector called Layer 00 [7]. The remaining seven
layers are radiation-hard, double sided detectors. The
first five layers after Layer 00 comprise the SVXII [8]
system and the two outer layers comprise the ISL [9] system. This entire system allows track reconstruction in
three dimensions. The resolution on the impact parameter for high-energy tracks with respect to the interaction
point is 40 µm, including a 30 µm contribution from the
beam-line. The resolution to determine z0 (z position
of the track at point of minimum distance to interaction vertex) is 70 µm. The 3.1 m long cylindrical drift
chamber (COT) [10] covers the radial range from 43 to
132 cm and provides 96 measurement layers, organized
into alternating axial and ±2o stereo superlayers. The
COT provides full coverage for |η| ≤1. The hit position
resolution is approximately 140 µm and the transverse
momentum resolution σ(pT )/p2T = 0.0015 GeV/c−1 .
Segmented electromagnetic and hadronic sampling
calorimeters surround the tracking system and measure
the energy flow of interacting particles in the pseudorapidity range |η| < 3.6. The central calorimeters (and
the end-wall hadronic calorimeter) cover the pseudorapidity range |η| < 1.1(1.3) and are segmented in towers
of 15o in azimuth and 0.1 in η. The central electromagnetic calorimeter [11] uses lead sheets interspersed with
polystyrene scintillator as the active medium and photomultipliers. The energy resolution
√ for high-energy electrons and photons is ≈ 13.5%/ E T ⊕2%, where the first
term is the stochastic resolution and the second term is
a constant term due to the non-uniform response of the
calorimeter. The central hadronic calorimeter [12] uses
steel absorber interspersed with acrylic scintillator as the
active medium.
The energy resolution for single-pions
√
is ≈ 75%/ E T ⊕3% as determined using the test-beam
data. The plug calorimeters cover the pseudorapidity
region 1.1 < |η| < 3.6 and are segmented in towers of
7.5o for |η| < 2.1 and 15o for |η| > 2.1. They are sampling scintillator calorimeters coupled with plastic fibers
and photomultipliers. The energy resolution of the plug
electromagnetic calorimeter
√ [13] for high-energy electrons
and photons is ≈ 16%/ E T ⊕1%. The energy resolution for √
single-pions in the plug hadronic calorimeter is
≈ 74%/ E T ⊕4%.
The collider luminosity is proportional to the average
number of inelastic pp̄ collisions per bunch crossing which
is measured using gas Čherenkov counters [14] located in
the 3.7 < |η| < 4.7 region.
The data selection (trigger) and data acquisition systems are designed to accommodate the high rates and
large data volume of Run II. Based on preliminary information from tracking, calorimetry, and muon systems,
the output of the first level of the trigger (level 1) is used
to limit the rate of the accepted events to ≈ 18 kHz at
the luminosity range 3→7 × 1031 cm−2 s−1 . At the next
trigger stage (level 2), with more refined information and
additional tracking information from the silicon detector,
the rate is reduced further to ≈ 500 Hz. The final level
of the trigger (level 3), with access to the complete event
information, uses software algorithms and a farm of computers to reduce the output rate to ≈ 100 Hz, which is the
rate at which events are written to permanent storage.
III.
DATA SAMPLE AND EVENT SELECTION
The expected signature of a tt̄ event in the all-hadronic
channel (tt̄ → bb̄ q1 q¯2 q3 q¯4 ) is the presence of six jets in
the reconstructed final state. Jets are identified as clusters of energy in the calorimeter using a fixed-cone algorithm with radius 0.4 in η-φ space [15]. The energy of the
jets needs to be corrected for various effects back to the
energy of the parent parton. The CDF jet energy corrections are divided into several levels to accommodate
different effects that can distort the measured jet energy:
non-uniform response of the calorimeter as a function of
η, different response of the calorimeter to different particles, non-linear response of the calorimeter to the particle
energies, uninstrumented regions of the detector, multiple pp̄ interactions, spectator particles, and energy radiated outside the jet clustering cone. In this analysis we
correct the energy of the jets taking into account all of
the above effects except those due to spectator particles
and energy radiated outside the cone. These additional
corrections are recovered using the transfer functions defined in Section IV.
A detailed explanation of the procedure to derive the
various individual levels of correction is described in
Ref. [16]. Briefly, the calorimeter tower energies are first
calibrated as follows. The scale of the electromagnetic
calorimeter is set using the peak of the dielectron mass
resonance resulting from the decays of the Z boson. For
the hadronic calorimeter we use the single pion test beam
6
data. This calibration is followed by a dijet balancing
procedure used to determine and correct for variations in
the calorimeter response to jets as a function of η. This
relative correction ranges from about -10% to +15%. After tuning the simulation to reflect the data, a sample of
simulated dijet events generated with pythia [17] is used
to determine the correction that brings the jet energies
to the most probable true in-cone hadronic energy. The
absolute correction varies between 10% and 40%.
A systematic uncertainty on these corrections is derived in each case. Some are in the form of uncertainties
on the energy measurement themselves, and some are
uncertainties on the detector simulation. Typical overall uncertainty is in the range of 3% to 4% for jets with
transverse momentum larger than 40 GeV/c. More details on the estimation of these uncertainties can be found
in [16].
The data sample is selected using a dedicated multijet trigger defined as follows. For triggering purposes the
calorimeter granularity is simplified to a 24 × 24 grid in
η-φ space. A trigger tower spans approximately 15o in φ
and 0.2 in η covering one or two physical towers. At level
1, we require at least one trigger tower with transverse
energy ETtow ≥ 10 GeV. At level 2, we require thePsum of
the transverse energies of all the trigger towers,
ETtow ,
be ≥ 175 GeV and the presence of at least four clusters
of trigger towers with ETcls ≥ 15 GeV. Finally, at level 3
we require four or more reconstructed jets with ET ≥ 10
GeV. This trigger selects about 80% of the tt̄ events in
the all-hadronic channel. The main background present
in this data sample is due to the production of multi-jets
via QCD couplings.
This analysis relies on Monte Carlo event generation
and detector simulation to model the tt̄ events. We
use herwig v6.505 [18] for the event generation. The
CDF II detector simulation [19] reproduces the response
of the detector to particles produced in pp̄ collisions.
Tracking of particles through matter is performed with
geant3 [20]. Charge deposition in the silicon detectors is calculated using a parametric model tuned to
the existing data. The drift model for the COT uses
the garfield package [21], with the default parameters
tuned to match COT data. The calorimeter simulation
uses the gflash [22] parameterization package interfaced
with geant3. The gflash parameters are tuned to testbeam data for electrons and pions. We describe the modeling of the background in Section V.
The events passing the trigger selection are further required to pass a set of clean-up cuts. First, we require
the reconstructed primary vertex [23] in the event to lie
inside the luminous region (|z| < 60 cm). In order to
reduce the contamination of the sample with events from
the leptonic tt̄ decays, we veto events which have a well
identified high-pT electron or muon [24], and require that
6ET
√P
be < 3 GeV1/2 [25], where the missing transverse
ET
energy, 6ET [26], is corrected for both the momentum of
any reconstructed muon and the position of the pp̄ in-
P
teraction point. The quantity
ET is the sum of the
transverse energies of jets.
After this preselection, we consider events with exactly
six jets, each with transverse energy ET ≥ 15 GeV and
|η| < 2. With these six jets, we calculate four variables that are used for the kinematic discrimination
of
P
tt̄ from background. One of theseP
variables is
ET defined above. Another variable,
3 ET , is the sum of
the transverse energies of jets removing
P the two leading
jets. We define centrality, C, as √ P 2ETP 2 , where
( E) −( pz )
P
P
E and
pz are the sum of the energies of jets and
the sum of the momenta of jets along the z-axis, respectively. The fourth variable is the aplanarity, A, defined
as 23 Q1 . Here Q1 is the smallest normalized eigenvalue
P j j
j
of the sphericity tensor Sab =
j Pa Pb , where Pa is
the momentum of a jet along one of the Cartesian axes.
We select eventsP
which satisfy the following kinematical
P
ET >
cuts: A + 0.005 3 ET > 0.96, C > 0.78, and
280 GeV. More details on the clean-up cuts, kinematical
and topological variables as well as the optimization of
the cuts are given in Ref. [27].
Since the final state of a tt̄ event is expected to contain
two jets originating from b quarks, their identification is
important for enhancing the tt̄ content of our final data
sample. Jets are identified as b jets using a displaced vertex tagging algorithm. This algorithm looks inside the
jet for good-quality tracks with hits in both the COT
and the silicon detector. When a displaced vertex can
be reconstructed from at least two of those tracks, the
signed distance (L2D ) between this vertex and the primary vertex along the jet direction in the plane transverse to the beams is calculated. The jet is considered
tagged if L2D /σ(L2D ) > 7.5, where σ(L2D ) is the uncertainty on L2D . This algorithm has an efficiency of
about 60% for tagging at least one b jet in a simulated
tt̄ event. More information concerning b tagging is available in Ref. [23]. In order to improve the signal purity,
we require the existence of at least one secondary vertex
tag in the event.
We introduce a new variable, minLKL, defined as the
minimum of the event probability calculated using the
matrix element technique (see Section IV for details).
Figure 1 shows the distribution of minLKL for a simulated tt̄ sample with Mtop = 175 GeV/c2 (continuous
line) and for the background (dashed line), after kinematical and b tagging requirements. Here and throughout this paper we use Mtop to label the top quark mass
used in the event generation. The event probability used
in Figure 1 is not normalized due to omission of multiplicative constants in its calculation. Although technically this variable is not a probability, we will keep using
this name. To further reduce the background contribution, we require that minLKL ≤ 10. The value of the
cut on minLKL is obtained by minimizing the statistical uncertainty on the top quark mass reconstructed using only the matrix element technique. The optimization
was done for various top mass quark values using a com-
7
Fraction of Events
bination of simulated tt̄ events and background events
(described in Section V).
0.14
Background
0.12
tt Simulation (Mtop=175 GeV/c2)
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
30
35
40
45
50
minLKL (Minimum of Negative Log Event Probability)
FIG. 1: Distribution of minLKL (minimum of the negative
log event probability) for simulated tt̄ events with Mtop =
175 GeV/c2 (continuous) and for background events (dashed)
modeled in Section V. The events pass the kinematical and
b-tagging requirements.
IV.
For each event passing our kinematical and topological requirements, we calculate the corresponding elementary cross section assuming tt̄ production followed by the
all-hadronic decay. In this calculation, we consider the
momentum 4-vectors of all the observed six jets, but we
assume them to be massless. The fraction of the total
tt̄ cross section contributed by an event can be interpreted as a probability density for the given event to be
part of the tt̄ production. As it is shown in Section IV A,
for each event this probability density depends on the top
quark mass. The mass value that maximizes the event
probability is used in the top quark mass reconstruction
technique described in Section VI. A likelihood function
obtained by combining the probability densities of a set
of events can also be used to reconstruct the top quark
mass. We use this technique in subsection IV B only to
validate the matrix element calculation used in the probability density determination, and not for the final mass
reconstruction.
A.
Table I shows the observed number of events in the
multi-jet data sample corresponding to a integrated luminosity of 943 pb−1 that pass the full event selection.
The table also shows the expected number of tt̄ events
(S) based on a sample of simulated tt̄ events assuming
the theoretical value of 6.7 pb [28] for the tt̄ production
cross section. The signal-to-background ratio (S/B) is
also shown, where the number of background events (B)
is taken as the difference between the observed number
of events and the tt̄ expectation (S). Based on the results reported in Table I, the minLKL cut improves the
signal-to-background ratio by a factor of three for the
sample where only one secondary vertex tag is required,
and by a factor of two for the sample where at least two
tags are required.
TABLE I: Number of observed multi-jet events passing the
event selection corresponding to an integrated luminosity of
943 pb−1 . The table also shows the expected number of
tt̄ events (S) and the corresponding signal-to-background ratio
(S/B). The number of tt̄ events is based on a sample of simulated tt̄ events assuming the theoretical value of 6.7 pb [28] for
the tt̄ production cross section. The number of background
events is taken as the difference between the observed number
of multi-jet events and the tt̄ expectation (S).
Selection
Single Tag
Double Tag
Observed S S/B Observed S S/B
Kinematical
782
71 1/10
148
47 1/2
minLKL ≤ 10
48
13 1/3
24
14 1/1
MATRIX ELEMENT TOOL
Definition of the probability density
For any event defined by a set of six 4-momenta, the
elementary cross section at a given top quark mass m can
be computed as if the event were the result of tt̄ production followed by all hadronic decay:
Z
dza dzb f (za )f (zb )
dσ(m, j) =
|M(m, j)|2
4Ea Eb |va − vb |
6 Y
d3~ji
4 (4)
×(2π) δ (EF − EI )
(1)
(2π)3 2Ei
i=1
Here, za (zb ) is the fraction of the proton(antiproton) momentum carried by the colliding partons; f (za ) and f (zb )
are the parton distribution functions for protons and for
antiprotons respectively; va and vb , and Ea and Eb represent the velocities and, respectively, the energies of the
colliding partons; j is a generic notation for all six 4momenta in the event assuming perfect parton identification and reconstruction; M(m, j) is the matrix element corresponding to tt̄ production and decay in the all
hadronic channel; EF (EI ) is a generic notation for the
4-vector of the final(initial) state.
If we sum the elementary cross sections of all the
events passing our event selection (trigger, kinematical
and topological) without the minLKL requirement, we
obtain a fraction (ǫ(m)) of the total tt̄ cross section,
σtot (m):
Z
σ(m) = dσ(m, j) = σtot (m)ǫ(m)
(2)
The fraction ǫ(m) is equivalent to the event selection efficiency for tt̄ events and is determined using samples of
8
simulated tt̄ events.
For each event, we define the probability density
P (j|m) as
P (j|m) =
dσ(m, j)
Q6
σtot (m)ǫ(m) i=1 d3~ji
Q6
(3)
3~
The quantity P (j|m) i=1 d ji is the probability for an
event defined by the set of six jets (i.e., six 4-momenta) to
be the result of tt̄ production followed by an all hadronic
decay for top quark mass m.
The final state partons from tt̄ decay are observed as
jets in our detector. Using simulated tt̄ events we calculate transfer functions, T F (~j|~
p), which represent a probability for a parton with momentum ~
p to be observed
as a jet with momentum ~j. The transfer functions are
described in Section IV A 3.
In order to enhance the features of the tt̄ phase space,
an additional weight, PT (~
p), is introduced. As it is shown
in Section IV A 4, this weight is obtained from the distribution of the transverse momentum of the tt̄ system in
simulated tt̄ events.
We assume that all six jets present in an all-hadronic
tt̄ event are the result of the hadronization of quarks in
the final state. There is an ambiguity in assigning the
jets to the quarks, and therefore all the possible combinations are considered and averaged. The counting of all
possible assignments is detailed in Section IV A 2. The
full expression of the probability density is given by:
6 X Z dza dzb f (za )f (zb ) Y
d3 p~i
P (j|m) =
4Ea Eb |va − vb |
(2π)3 2Ei
combi
i=1
|M(m, p)|2 (2π)4 δ (4) (EF − EI )T F (~j|~
p)PT (~
p)
(4)
×
σtot (m)ǫ(m)Ncombi
where the sum is performed over all jet-parton combinations and Ncombi is the total number of possible jetparton assignments.
The calculation of the matrix element M(m, p) is detailed in Section IV A 1 and the integration performed in
Eq. 4 is described in Section IV A 5.
1.
Calculation of the matrix element
Two processes contribute to tt̄ production: gluongluon fusion and quark-antiquark annihilation. At the
Tevatron, about (15 ± 5)% of tt̄ events are expected to
be produced by gluon-gluon fusion while the remaining
85% are produced by quark-antiquark annihilation [28].
In addition, 90% of the simulated tt̄ events produced by
quark-antiquark annihilation result from uū annihilation.
Given that having both types of tt̄ production doubles the
calculation time, we only use the matrix element describ¯ b̄ūd). To validate this
ing the process uū →tt̄ → (bud)(
choice, we reconstruct the top quark mass using a uū matrix element in a sample of tt̄ events produced only via
gluon-gluon fusion. We observe a negligible bias (0.0 ±
0.2 GeV/c2 ) in the reconstruction of the top quark mass
and we conclude that using a matrix element with uū as
the initial state should be sufficient for the mass reconstruction.
For the final state, having a W boson decay into a
ud¯ pair or a cs̄ pair results in no difference for the final
reconstruction as both pairs of quarks will be observed
as jets. The other hadronic decays are suppressed since
their rate is proportional to the square of small elements
of the Cabibbo-Kobayashi-Maskawa matrix [29].
In the high-energy limit (or the massless limit), the
solutions to the Dirac equation can be written as
1
p
(1 − p̂ · ~σ )ξ
2
u(p) = 2Ep 1
(1 + p̂ · ~σ )ξ
21
p
(1 − p̂ · ~σ )η
2
(5)
v(p) = 2Ep
− 21 (1 + p̂ · ~σ )η
where p = (Ep , p~) is the 4-momentum of a particle.
The solution with positive frequencies is u(p), and that
with negative frequencies is v(p); σ µ = (1, ~σ ) and σ µ =
(1, −~σ), where ~σ are the Pauli spin matrices.
The presence of the operator p̂ · ~σ will project the spin
states along the direction of motion defined by p̂. For
a particle traveling in the direction defined by the polar
angle θ and by the azimuthal angle φ, the spin states
along this direction are given by Eq. 6.
cos θ2
−e−iφ sin 2θ
ξ(↑) =
,
ξ(↓)
=
(6)
eiφ sin θ2
cos θ2
For an antiparticle we have η(↑) = ξ(↓) and η(↓) =
−ξ(↑). Given these relations and assuming that the incoming partons travel along the z-axis, the matrix element has only two non-zero terms due to the initial state
partons, IRR and ILL . These are 4-vectors and correspond to the situations when the incoming partons have
the same chirality. Considering the proton going in the
positive direction, these terms are:
q
p
µ
IRR
=
2Euin 2Euin (0, 1, i, 0)
q
p
µ
ILL
=
(7)
2Euin 2Euin (0, 1, −i, 0)
where Euin and Euin are the energies of the incident u
quark and u quark, respectively.
Omitting all multiplicative constants, we express the
matrix element squared as
X
|M|2 = FE2 · Peg · Pet · Pet · PeW1
|M|2 →
spins
colors
×PeW2 · (|MRR |2 + |MLL |2 )
(8)
9
two secondary vertex tags, we assign to b quarks only
the two jets with the highest transverse energy. Note
that the quarks in the decay of either W boson can not
be interchanged in the matrix element calculation as one
is particle and the other is antiparticle and they have
different spinors.
where the factors entering this expression are

q


FE = (Eb )(Eb )(Eu )(Ed )(Ed )(Eu )(Euin )(Euin )




1
2

 Peg = |Pg | = (pu +pu )4


 Pe =
1
2
2 2
2 2
t,t
(pt,t −m ) +m Γt

1

PeW1,2 = (P 2
2 2
2
2


W+,− −MW ) +MW ΓW




†


M
=
S
ξb (↓) (W− · σ) 0
I
 I
0
(W+ · σ) ξb† (↓)
(9)
In Eq. 9 MW and ΓW are the mass and the width of the
W boson, m and Γt are the mass and the width of the
top quark, and W+ and W− are the 4-momenta of the
W bosons.
Also in Eq. 9 MI stands for both MRR and MLL
(Eq. 8), the difference arising from the definition of the
symbol SI . The symbol SI is defined as
SI = pµt γµ I ν γν ptρ γρ + m2 I µ γµ
(10)
where γµ are the Dirac matrices and I is either IRR or
ILL . We calculate MRR and MLL using Eq. 6 and matrix
algebra [30].
2.
Combinatorics
While there are 6! = 720 ways to assign the observed
jets to the six partons of the final state in all hadronic
tt̄ decay, we can take into account a reduced number of
possibilities by making a few observations and assumptions.
In the case of the process uū → tt̄ , assuming that
the masses of the up quarks are negligible and omitting
the constant and the gluon propagator terms, the spin
averaged matrix element squared is
1 X
|M|2 ≈ (pu ·pt )(pu ·pt )+(pu ·pt )(pu ·pt )+m(pu ·pu )
4 spins
(11)
where p is the 4-momentum of a particle.
From this expression, the t ↔ t symmetry is evident.
The symmetry holds also for the matrix element of the
process containing the decay of the top quarks. This symmetry can be translated into a symmetry to b ↔ b once
we consider all possible b-W pairings for each top quark:
{t = (b1 , W1 ), t = (b2 , W2 )}, {t = (b1 , W2 ), t = (b2 , W1 )}.
It is obvious that swapping the b’s is equivalent with
swapping the top quarks.
In conclusion, due to the t ↔ t symmetry the number
of relevant combinations is 360. Secondly, if any of the
jets is identified as a secondary vertex tag we assume
that jet be produced by a b quark. This assumption
results in a factor of three reduction of the number of
relevant combinations, down to 120 (or 5!). If there is
an additional secondary vertex tag, we get a factor of
five reduction down to 24 (or 4!). If there are more than
3.
Transfer functions
The transfer functions, T F (~j|~
p), express the probability for a parton with momentum p to be associated with
a jet reconstructed to have momentum j. The transfer
function term from Eq. 4 is in fact a product of six terms,
one for each of the final state quarks: two for the b quarks
and four for the decay products of the W bosons. For
each jet in the final state we assume that the jet axis is
the same as that of the parton that went on to form the
jet. Making the change of variables j → ζ = 1 − j/p, the
expression for T F (~j|~
p) becomes:
T F (~j|~
p) →
6
Y
i=1
T F (j~i |~
pi ) =
6
Y
i=1
g
T
F (ζ(ji )|pi )
(−1) (2)
δ (ΩJi − ΩPi )
×
pi
(12)
where ΩJi and ΩPi are the solid angles of the jets
and of the quarks, respectively. The transfer functions
g
T
F (ζi |pi ) are built using simulated tt̄ events with Mtop =
175 GeV/c2 surviving the trigger, kinematical and topological requirements. The choice of Mtop = 175 GeV/c2
is arbitrary as our studies show that the transfer functions have a negligible dependence on the mass of the top
quark in the range 150 GeV/c2 to 200 GeV/c2 . In this
sample, we associate a jet with p
a parton if their separation in the η − φ space is ∆R = ∆η 2 + ∆φ2 ≤ 0.4. We
define a jet to be matched to a parton if no other jet satisfies this geometrical requirement. We define a tt̄ event
to be matched if each of the six partons in the final state
has a unique jet matched to it. The transfer functions
are built out of the sample of matched events.
The jets formed by partons from W -bosons decays have
a different energy spectrum from that of the jets originating from the b quarks. Thus we form different sets of
transfer functions depending on the flavor of the parton
the jet has been matched to.
The transfer functions are described using a parameterization in bins of the parton energies and of the parton
pseudorapidities. We use three bins for the pseudorapidity: 0 ↔ 0.7, 0.7 ↔ 1.3, and 1.3 ↔ 2.0. Table II shows
the definition of energy binning for the b-jet transfer functions, while Table III is for the W -jet transfer functions.
The energy binning is chosen such that the distributions
for transfer functions are smooth. In each bin, the shape
of the transfer function is fitted to a normalized sum of
two Gaussians.
10
Bin
1
2
3
4
5
6
7
0 ≤ |η| < 0.7
10 → 53
53 → 64
64 → 74
74 → 85
85 → 97
97 → 114
114 → ∞
0.7 ≤ |η| < 1.3
10 → 83
83 → 111
111 → ∞
1.3 ≤ |η| ≤ 2.0
10 → ∞
Using the same simulated tt̄ events as for transfer functions, in Fig. 2 we show the distribution of the magnitude
pTtt̄ of the transverse momentum of the tt̄ system. A sum
of three Gaussians is a good fit of this distribution.
Events/ 1 GeV/c
TABLE II: Definition of the binning of the parton energy
for the b-jet transfer functions parameterization for various η
bins. The unit for the energy values is GeV.
18000
16000
14000
12000
10000
8000
6000
4000
2000
TABLE III: Definition of the binning of the parton energy for
the W -jet transfer functions parameterization for various η
bins. The unit for the energy values is GeV.
Bin
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 ≤ |η| < 0.7
10 → 32
32 → 38
38 → 44
44 → 49
49 → 54
54 → 59
59 → 64
64 → 69
69 → 75
75 → 81
81 → 89
89 → 99
99 → 113
113 → ∞
4.
0.7 ≤ |η| < 1.3
10 → 50
50 → 63
63 → 76
76 → 90
90 → 108
108 → ∞
1.3 ≤ |η| ≤ 2.0
10 → 98
98 → ∞
Transverse momentum of the tt̄ system
The PT (~
p) weight (introduced in Eq. 4) is a function
dependent on the momenta of the partons in the final
state, generically represented by p~ in the argument of
the function. More exactly, this weight depends on the
magnitude of the transverse momentum of the tt̄ system,
pTtt̄ , and azimuthal angle, φTtt̄ . As we expect to have a
flat dependence on φTtt̄ we express this through a factor
of 1/2π. We define the function PeT (pTtt̄ ) to express the
dependence on pTtt̄ . We write in Eq. 13 the expression
of the weight due to the transverse momentum of the
tt̄ system.
q
PeT pTtt̄ = (pxtt̄ )2 + (pytt̄ )2
q
PT (~
p) → PT (pxtt̄ , pytt̄ ) =
2π (pxtt̄ )2 + (pytt̄ )2
(13)
where pTtt̄ is shown in its Cartesian form using the projections of the transverse momentum of the tt̄ system along
the x and y axes.
0
0
20
40
60
80
100
120
140
160
180
200
PT of tt (GeV/c)
FIG. 2: Magnitude of the transverse momentum of the tt̄ system in simulated tt̄ events. The fit is a sum of three Gaussians.
5.
Implementation and evaluation of the probability density
The Sections IV A 2, IV A 1, IV A 3 and IV A 4
present details on the expressions of several important
pieces entering the probability density. To carry out the
integration over parton momenta, we change to a spherical coordinate system. The delta functions δ (2) (ΩJi −
ΩPi ) present in the expression of the transfer functions
T F (~j|~
p)(Eq. 12) allow us to drop all integrals over the
parton angles.
To further reduce the number of integrals we use the
narrow width approximation for the W bosons. This
results in two more delta functions for the squares of the
propagators of the two W bosons exemplified in Eq. 14
for both bosons.
PeW =
1
2
(PW
−
2 )2
MW
ΓW ≪MW
+
2 Γ2
MW
W
2
2
−→ δ(PW
− MW
)
−→
π
M W ΓW
(14)
In the high-energy limit, the invariant mass of the W + boson decay products is given by:
2
PW
+ = 2p1 p2 sin θ1 sin θ2 (cosh ∆η12 − cos ∆φ12 ) =
= 2p1 p2 ω12 (Ω1 , Ω2 )(15)
where ∆η12 is the difference in pseudorapidities of the
two decay partons and ∆φ12 = π − ||φ1 − φ2 | − π| is the
difference between their azimuthal angles.
2
Making the change of variables PW
+ → p1 , Eq. 14 can
be written as:
PeW +
ΓW ≪MW
−→
1
π
δ(p1 − p01 ) (16)
MW ΓW 2p2 ω12 (Ω1 , Ω2 )
11
As described in section IV A 1, we assume that the incoming partons have zero transverse momentum. This
would, in principle, result in violation of momentum conservation in the transverse plane as we consider non-zero
transverse momentum for the tt̄ system in the ME calculation. However, we expect this to be a small effect
covered by the uncertainty on the parton distribution
functions of the proton and of the antiproton. We can
omit the delta functions requiring energy conservation
along the x and y axes, resulting in
δ
(4)
(EF − EI ) → δ Ea + Eb −
×δ pza + pzb −
= δ pu + pu −
6
X
i=1
pi
i=1
6
X
i=1
6
X
pzi =
6
X
pi δ pu − pu −
pzi
(17)
i=1
We make the change of variables za → pu and zb →
pu since za = pu /pproton and zb = pu /pantiproton . The
values of the proton and antiproton momenta, pproton
and pantiproton , are constant and from now on we drop
them from any expressions. In the high-energy limit we
have |va − vb | = 2c and we omit this term since c is
a constant, the speed of light. We express the energyconserving delta function as
δ (4) (EF − EI ) → δ pu + pu −
×δ pu − pu −
6
X
i=1
6
X
i=1
pi
pi cos θi =
1
(18)
= δ(pu − p0u )δ(pu − pu0 )
2
P6
P6
where p0u = i=1 pi (1 + cos θi )/2 and pu0 = i=1 pi (1 −
cos θi )/2.
In section IV A 4, we expressed PT (~
p) as a function
of the projections of the transverse momentum of the
tt̄ system along the x and y axes (Eq. 13). We will make
a change of variable from the b-quark momenta to these
variables. The Jacobian of this transformation
1
sin θb sin θb (cos φb sin φb − sin φb cos φb )
(19)
is obtained by solving the system of equations for pb and
J(b → 6) =
pb .
P
pxtt̄ = pb cos φb sin θb + pb cos φb sin θb + 6i=3 pxi
P
6
pytt̄ = pb sin φb sin θb + pb sin φb sin θb + i=3 pyi
(20)
We write the expression of the probability density in
its final form as
X Z
dpxtt̄ dpytt̄ dp2 dp4
P (j|m) =
σtot (m)ǫ(m)Ncombi
combi
6 J(b → 6)pb pb f (p0u )f (pu0 ) Y g
×
T F (ζi |pi )
(ω12 )2 (ω34 )2 p2 p4
i=1
×
PeT (pTtt̄ ) e e e
· Pg · Pt · Pt · (|MRR |2 + |MLL |2 )
pTtt̄
(21)
We evaluate the integrals in Eq. 21 numerically. The integration is performed in the interval [−60, 60] GeV/c for
tt̄
the variables px,y
and [10, 300] GeV/c for the variables
p2,4 . The step of integration is 2 GeV/c. Based on a
sample of tt̄ events where Mtop = 175 GeV/c2 passing
the event selection, we choose these integration ranges
such that the distributions of the parton level variables
tt̄
(px,y
and momenta of W -boson decay partons) are contained well (99%) within them. Given these limits, at
each step of integration we have to make sure that all
momenta entering Eq. 21 have positive magnitudes. The
probability density is evaluated for top mass values going
in 1 GeV/c2 increments from 125 GeV/c2 to 225 GeV/c2 .
The dependence on mass of the tt cross section is obtained from values calculated at leading-order by comphep [32] Monte Carlo generator for the processes uu →
tt, dd → tt and gg → tt. The absolute values for these
cross sections are not as important as their top mass dependence, which is shown in Fig. 3.
tt cross-section (pb)
2
where p01 = MW
/(2p2 ω12 ). In the case of the W − boson we use equations similar to Eqs. 15 and 16, but with
2
different notations: the change of variables is PW
− → p3
0
2
and the pole of the delta function is p3 = MW /(2p4 ω34 ).
The mass and width of the W boson are fixed at
80.4 GeV/c2 and 2.1 GeV/c2 , respectively [31].
25
20
15
10
5
0
120
140
160
180
200
220
2
Top Quark Mass (GeV/c )
FIG. 3: Cross section for tt production as a function of the
top quark mass, as obtained from comphep [32].
For the proton and antiproton parton distribution
functions (PDF), f (p0u )f (pu0 ), we use the cteq5l [33]
distributions with the scale corresponding to a top mass
of 175 GeV/c2 . The tt acceptance, ǫ(m), is described in
which is expected to have a maximum around the true
top quark mass of the sample. Finding the value of the
top quark mass that maximizes the likelihood represents
the traditional method for reconstructing the top quark
mass using a matrix element technique [4]. However,
we use this reconstruction technique only to check the
matrix element calculation.
We use the simulated tt̄ samples generated with various top quark masses. For each sample, we reconstruct
the top quark mass using the traditional matrix element
technique and compare the reconstructed mass to the
true input mass Mtop for several different input mass values. Ideally, we should see a linear dependence with no
bias and a unit slope.
The first check is done at the parton level. We smear
the energies of the final state partons from our simulated tt̄ events and use these numbers to describe the
jets. The parton energies are smeared according to the
transfer functions described in section IV A 3. Figure 4
shows the linearity check in this case. We observe a slope
of ≈1 and a bias of 0.9 GeV/c2 .
We perform the same test using the energies of the
jets matched to the partons. Figure 5 shows the linearity
check. Here the bias is 1.2 GeV/c2 , but the slope remains
≈1. The final test we perform to validate the matrix
element calculation uses fully reconstructed signal events
where we allow events to include mismatched jets as well.
Figure 6 shows the linearity check in this case. The bias
is no longer the same for all masses as the slope is 0.94
± 0.01.
Although there is some bias, all checks we list above
show the good performance of our matrix element calculation. In general, the traditional matrix element approach [3] is expected to provide a better statistical uncertainty on the top mass than the template analyses [5].
In the case of the present analysis, our studies show that
the traditional matrix element method does better only
when the mass reconstruction is performed on signal samples. When the background is mixed in, the template
method we use has a greater sensitivity and by construction eliminates the bias of the matrix element calculation
(see Section VI).
180
170
y=x
160
y = p0 + (x - 178)p1
p0 = 177.1 ± 0.1
p1 = 0.99 ± 0.01
150
150
160
170
180
190
200
2
FIG. 4: Reconstructed top mass versus input top mass at
parton level. The energies of the partons have been smeared
using the transfer functions. The continuous line y = x is
added for visual reference.
2
events
190
Top Quark Mass Mtop (GeV/c )
Reconstructed Top Mass (GeV/c )
The event probability described in the Section IV A is
expected to have a maximum around the true top quark
mass in the event. Multiplying all the event probabilities
we obtain a likelihood function,
Y
L(Mtop ) =
P (j|Mtop )
(22)
200
200
190
180
170
y=x
160
y = p0 + (x - 178)p1
p0 = 179.2 ± 0.1
p1 = 1.01 ± 0.01
150
150
160
170
180
190
200
2
Top Quark Mass Mtop (GeV/c )
FIG. 5: Reconstructed top mass versus input top mass using
jets that were uniquely matched to partons. The continuous
line y = x is added for visual reference.
2
Validation of the matrix element calculation
Reconstructed Top Mass (GeV/c )
B.
2
Section IV A.
Reconstructed Top Mass (GeV/c )
12
200
190
180
170
y=x
160
y = p0 + (x - 178)p1
p0 = 178.5 ± 0.1
p1 = 0.94 ± 0.01
150
150
160
170
180
190
200
2
Top Quark Mass Mtop (GeV/c )
FIG. 6: Reconstructed top mass versus input top mass using
realistic jets. The continuous line y = x is added for visual
reference.
13
Tagging matrix
The tagging matrix is a parameterization of the heavy
flavor rates as a function of the transverse energy of jets,
the number of tracks associated to the jet and the number
of primary vertices in the event. Using the b-tagging algorithm described in Section III, we determine the above
rates in a sample (4J) largely dominated by QCD multijet processes and selected from multi-jet data events with
exactly four jets and passing the clean-up requirements
described in Section III.
We use a control region to check our assumption that
the tagging rates from the 4-jet sample can be used to
predict the tagging rates as a function of the variables
used in the kinematical selection. This control region
(CR1) contains events with exactly six jets and passing
the clean-up cuts. The signal-to-background ratio in this
region is about 1/250, estimated using same method as
for the BG sample. We compare the observed rates with
the predicted rates based on the tagging matrix. Figure 7 shows the comparison for events with exactly one
secondary vertex tag, while Fig. 8 shows the comparison
in the sample with at least two secondary vertex tags.
The variables chosen for this comparison are the transverse energies of jets, sum of the transverse energies of
the six leading jets, aplanarity, and centrality as defined
in Section III. The Kolmogorov-Smirnov probabilities for
these comparisons in the single (double) tagged samples
are: 0.0 (8.6E-5), 3E-11 (0.69), 0.99 (4.3E-3), and 0.12
(0.05), respectively.
Based on Fig. 7(a), the discrepancy between the observed rate and the predicted rate for jets with low transverse momentum may be an artifact of the binning of the
tagging matrix. For transverse energies between 15 GeV
Fraction/ 6GeV/c2
A.
(a)
0.1
0.08
0.06
0.04
0.02
0
0
50
100
150 2 200
Jet Et (GeV/c )
0.2
(c)
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
Aplanarity (A)
0.5
0.05
(b)
0.04
0.03
0.02
0.01
0
0
200
400
600
2
Sum of Transverse Energy (GeV/c )
250
Fraction/ 0.02 C
In this section we describe the data-driven technique
used to model the background for this analysis. The
technique uses jet energies which are measured in the
calorimeter and so are unchanged by jet energy scale
changes. Properties of the model are checked by comparison with a simulated sample of events containing the
final state bb + 4 light partons.
The modeling of background is based on a subset of
the multi-jet data sample depleted of tt̄ events where the
heavy flavor jets are identified according to backgroundlike heavy flavor rates (tagging matrix), described in Section V A. The subset of multi-jet data is selected applying the event selection of Section III excluding the
minLKL and the secondary vertex tag requirements.
This sample (BG) counts 2652 events, with an estimated
signal-to-background ratio of about 1/25. For this ratio we estimate the signal from a sample of simulated
tt̄ events assuming a tt̄ production cross section of 6.7 pb.
The estimate for the background is equal to the number
of observed events in the BG sample.
and 40 GeV the tagging matrix uses the average rate,
and therefore the rates for smaller intervals in this range
might not be predicted well. Figures 7(a) and 8(a) support this by showing that, for this range of transverse
energies, half of the data points are below and the other
half is above the solid histogram representing our background model.
The overall agreement between the observed and predicted rates is quite poor. In principle, a systematic uncertainty should be assigned to cover this discrepancies.
However, the templates used in the mass measurement
use the event probability based on matrix element information and they will be less affected by these inaccuracies. The reason for this is the fact that we use only
a tt̄ matrix element. For background events the event
probability (Eq. 21)is flat as a function of the assumed
top quark mass. The flatness of the event probability
results in wide templates for the background sample and
the systematic effects due to the mistag matrix will get
smeared. In fact, the background templates in the control regions defined in Section V B agree very well with
the corresponding distributions based on the simulation
of background events with bb + 4 light partons in the
final state.
We conclude that the tagging matrix can be used to
predict the background-like heavy flavor rates for events
with same jet multiplicity as expected for the all-hadronic
tt̄ events. More details on the tagging matrix can be
found in Ref. [27].
Fraction/ 5 GeV/c2
BACKGROUND MODEL
Fraction/ 0.02 A
V.
0.06
(d)
0.04
0.02
0
0
0.2
0.4
0.6
Centrality (C)
0.8
1
FIG. 7: Background validation in control region CR1 for single tagged events from the multi-jet data (dots) and from the
background model (solid histogram). The distributions are
normalized to the same area.
B.
Estimation of the background
Based on the tagging matrix, a jet has a certain probability (rate) to be tagged as a heavy flavor jet depending on its transverse energy, number of tracks associated
to it and number of vertices in the event. For a jet
with a transverse energy between 15 GeV and 40 GeV
Fraction/ 6GeV/c2
0.12
0.1
0.08
0.06
0.04
0.02
0
0
(a)
50
0.2
100
150 2 200
Jet Et (GeV/c )
(c)
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
Aplanarity (A)
0.5
0.05
(b)
0.04
0.03
0.02
0.01
0
0
200
400
600
2
Sum of Transverse Energy (GeV/c )
250
Fraction/ 0.02 C
Fraction/ 0.02 A
Fraction/ 5 GeV/c2
14
(d)
0.06
TABLE IV: Definition of the control regions used in the background modeling procedure. The selection requirements that
differentiate them are defined in Section III.
0.04
0.02
0
0
contains events that pass all our selection requirements
without the minLKL cut and has a signal-to-background
ratio of about 1/6. The signal region SR has events passing all selection criteria defined in Section III. Table IV
summarizes all the regions used in our background modeling procedure.
0.2
0.4
0.6
Centrality (C)
0.8
1
FIG. 8: Background validation in control region CR1 for double tagged events from the multi-jet data (dots) and from the
background model (solid histogram). The distributions are
normalized to the same area.
and with ten associated tracks, this probability is (7.2
± 0.5)%. Using these probabilities we tag the jets as
originating from a b quark. This tagging procedure is
repeated 20,000 times in the events of the BG sample
producing about ten million tagged configurations which
are interpreted as background events.
A tagged configuration is an event from the BG sample where at least one of the six jets is tagged using the
tagging matrix. Such kind of event can produce many
tagged configurations which are unique if they have different tagged jets or a different number of tagged jets. We
find 12888 unique single tagged configurations, and 26715
unique double tagged configurations. Of these, 657 (or ≈
(5.1 ± 0.2)%) single tagged configurations and 1180 (or
≈ (4.4 ± 0.1)%) double tagged configurations pass the
minLKL cut. We use these configurations, unique or
duplicate, to form all relevant background distributions
used for various checks and for the final measurement.
The estimated number of background events is defined
as the difference between the total number of events observed in the data sample and the expected number of
tt̄ events based on the standard model expectation for
tt̄ production cross section of 6.7 pb [28]. This normalization applies to the top quark mass reconstruction procedure described in Section VI, and for the validation of
the background model described below.
We check various distributions of the background
events modeled above against those from a sample of
simulated events with bb + 4 light partons in the final
state. This simulated sample is built using alpgen [35]
for the event generation, pythia for the parton showering, and the detector simulation as described in Section III. Given our event selection, other background
sources are expected to have smaller contributions compared to the one from bb + 4 light partons and therefore
affect less the relevant distributions.
This check is performed in a control region (CR2) and
in the signal region (SR) defined as follows. Region CR2
Region Clean-up Njets Kinem. minLKL b-tag Nevents
4J
yes
4
no
no
no 2,242,512
BG
yes
6
yes
no
no
2652
CR1
yes
6
no
no
no 380,676
CR2
yes
6
yes
no
yes
930
SR
yes
6
yes
yes
yes
72
Given that the BG sample used in our background
model contains a small tt̄ content, we need to correct
all the background distributions built from it. The relationship between a given uncorrected background distribution, fB , and the corrected one, fBcorr is
fBcorr =
f B − aS f S
1 − aS
(23)
where fS is the corresponding distribution for tt̄ events
and aS is the fraction of the uncorrected background sample due to tt̄ events. These quantities for tt̄ are determined from a sample of simulated tt̄ events where Mtop
= 170 GeV/c2 by randomly tagging the jets using the
tagging matrix defined in Section V A. We choose the
above value for the top quark mass based on the value of
the world mass average [34] at the time of this analysis;
in Section VIII we determine a systematic uncertainty
due to this choice. The expression for aS in region CR2
is
=
aCR2
S
NSCR2
B CR2 + NSCR2
(24)
where B CR2 is the background estimate in this region
and NSCR2 is the number of tt̄ events estimated using the
tagging matrix. The expression for aS in region SR is
aSR
S =
NSCR2 ǫminLKL
S
B CR2 ǫminLKL
+ NSCR2 ǫminLKL
B
S
(25)
where ǫminLKL
(ǫminLKL
) is the efficiency of the
S
B
minLKL cut for tt̄ (background) in the CR2 region. The
efficiency for background is determined using the ratio of
the number of uniquely tagged configurations before the
minLKL cut (12,888 single tagged and 26,715 double
tagged), and after the minLKL cut respectively (657 single tagged and 1180 double tagged). Table V shows the
estimated number of background events B CR2 and the
efficiency of the minLKL cut for background ǫminLKL
B
15
in region CR2. Tables VI and VII show the values for
ǫminLKL
, NSCR2 , and aCR2
in region CR2 as well as the
S
S
values of aSR
S for simulated tt̄ samples with different values on Mtop .
TABLE V: The estimated number of background events
B CR2 and the efficiency of the minLKL cut for background
ǫminLKL
in region CR2. The number of background events is
B
the difference between the observed number of events and the
expected number of tt̄ events assuming a tt̄ production cross
section of 6.7 pb.
Parameter
B CR2
minLKL
ǫB
Single Tag
711
0.051
Double Tag
101
0.044
(Fig. 9). The values of the Kolmogorov-Smirnov probabilities are 25% for the samples with single tagged events,
and 43% for the samples with double tagged events. For
the signal region, we look at the invariant mass of all the
untagged pairs of jets in the event (Fig. 10) and at the
most probable per-event top quark mass (Fig. 11). These
are variables of particular interest in this region as they
will be used in the reconstruction of the top quark mass
and for the in situ calibration of the jet energy scale,
as described in Section VI. Based on the comparison
from Fig. 10, the Kolmogorov-Smirnov probabilities are
90% for the single tagged events and 70% for the double
tagged events.
TABLE VI: The number of tt̄ events, NSCR2 , with one jet identified as b jets using the tagging matrix; in region CR2, the
acceptance of the minLKL cut for tt̄ events, ǫminLKL
, and
S
the values of the parameters aCR2
(Eq. 24), and aSR
S
S (Eq. 25)
for simulated tt̄ samples with different values on Mtop .
NSCR2
29
30
28
28
ǫminLKL
S
0.21
0.20
0.19
0.18
aCR2
S
0.039
0.040
0.038
0.038
NSCR2 ,
TABLE VII: The number of tt̄ events,
with at least
two jets identified as b jets using the tagging matrix; in region CR2, the acceptance of the minLKL cut for tt̄ events,
ǫminLKL
, and the values of the parameters aCR2
(Eq. 24), and
S
S
SR
aS (Eq. 25) for simulated tt̄ samples with different values on
Mtop .
Mtop (GeV/c2 )
160
170
175
180
NSCR2
2
2
2
2
ǫminLKL
S
0.31
0.29
0.29
0.27
aCR2
S
0.019
0.019
0.019
0.019
aSR
S
0.133
0.126
0.126
0.118
The correction procedure uses by default the parameters as derived for Mtop = 170 GeV/c2 . In the determination of the systematic uncertainty due to this choice, we
use the parameters corresponding to Mtop = 160 GeV/c2 ,
and Mtop = 180 GeV/c2 , respectively (see Section VIII).
The parameters obtained using Mtop = 175 GeV/c2 are
given for reference in Table VI as that mass value corresponds to a tt̄ production cross section of 6.7 pb.
Following this correction procedure, we compare
shapes between our background model and the sample of
simulated bb + 4 light partons described above. First, we
do this comparison in region CR2 where we look at the invariant mass of all the untagged pairs of jets in the event
Background with 1 b-tag
0.12
Model
Monte Carlo
0.1
0.08
0.06
0.04
aSR
S
0.146
0.144
0.130
0.124
0.02
0
0
100
200
300
2
Dijet Mass (GeV/c )
(b)
Fraction/ 14GeV/c2
Mtop (GeV/c2 )
160
170
175
180
Fraction/ 14GeV/c2
(a)
Background with 2 b-tags
0.14
Model
0.12
Monte Carlo
0.1
0.08
0.06
0.04
0.02
0
0
100
200
300
2
Dijet Mass (GeV/c )
FIG. 9: Invariant mass of pairs of untagged jets in control
region CR2 for alpgen bb + 4 light partons (cross), and for
the background model (solid): (a) for single tagged events
(Kolmogorov-Smirnov probability is 25%) and (b) for double
tagged events (Kolmogorov-Smirnov probability is 43%).
These comparisons show good agreement between our
data-driven background model and a simulated sample
of events containing the final state bb + 4 light partons,
obtained using the alpgen generator. In Section VIII we
evaluate the effect on the reconstructed top quark mass
due to the limited statistics available in sample BG to
construct the background model.
16
(a)
Fraction/ GeV/c2
Fraction/ 14GeV/c2
(a)
Background with 1 b-tag
0.25
Model
Monte Carlo
0.2
0.15
0.1
Background with 1 b-tag
Model
0.25
Monte Carlo
0.2
0.15
0.1
0.05
0
0
0.3
0.05
100
200
300
0
2
140
160
180
Dijet Mass (GeV/c )
220
2
(b)
Fraction/ GeV/c2
(b)
Fraction/ 14GeV/c2
200
Event Top Mass (GeV/c )
Background with 2 b-tags
0.45
Model
0.4
Monte Carlo
0.35
0.3
0.25
Background with 2 b-tags
0.3
Model
Monte Carlo
0.25
0.2
0.15
0.2
0.1
0.15
0.1
0.05
0.05
0
0
100
200
300
0
2
Dijet Mass (GeV/c )
FIG. 10: Invariant mass of pairs of untagged jets in signal region for alpgen bb + 4 light partons (cross), and for
the background model (solid): (a) for single tagged events
(Kolmogorov-Smirnov probability is 90%), and (b) for double
tagged events (Kolmogorov-Smirnov probability is 70%).
VI.
TOP QUARK MASS ESTIMATION
Our technique starts by modeling the data using a mixture of signal events obtained from tt̄ simulation and of
background events obtained via our background model.
The events are represented by two variables: the invariant mass of pairs of untagged jets and an event-by-event
reconstructed top mass described below. These two variables are used to form distributions (templates), separately for tt̄ events and for background events. In the
case of tt̄ events, the templates are parameterized as a
function of the mass of the top quark and the jet energy
scale (JES) variable (defined below). For background no
such dependences are expected since they contain no top
quark and the jet energies used for the background modeling are taken from data. The measured values for the
top quark mass and for the JES are determined using a
likelihood technique described in Section VI B.
The largest contribution to the systematic uncertainty
on the top quark mass is due to the uncertainty on the jet
energy scale. To limit the impact of this systematic on
the total uncertainty on the top quark mass, we use an in
situ calibration of the jet energy scale via the W -boson
140
160
180
200
220
2
Event Top Mass (GeV/c )
FIG. 11: Event by event most probable top quark masses in
the signal region for alpgen bb + 4 light partons (cross), and
for the background model (solid): (a) for single tagged events,
and (b) for double tagged events.
mass. We measure a parameter JES that represents a
shift in the jet energy scale from our default calibration
as defined in Section III. This quantity is expressed in
units of the total nominal jet energy scale uncertainty
σc that is derived following the default calibration. This
uncertainty depends on the transverse energy, pseudorapidity, and the electromagnetic fraction of the jet energy.
On average, the uncertainty is approximately equivalent
to a 3% change in the jet energy scale for jets in tt̄ events.
By definition, JES = 0 σc represents our default jet energy scale; JES = 1 σc corresponds to a shift in all jet
energies by one standard deviation; and so on.
The templates for tt̄ events are determined from samples of simulated tt̄ events with Mtop ranging from
150 GeV/c2 to 200 GeV/c2 in steps of 5 GeV/c2 . We
also include the sample where Mtop = 178 GeV/c2 for a
total of twelve different tt̄ simulated mass samples. In
addition to the variation of the top quark mass, for each
value of Mtop we consider seven values for JES between
-3 σc and 3 σc , in steps of 1 σc . We use the events obtained from our background model to form the templates
for the background.
A.
Definition and parameterization of the
templates
where the parameters αi depend on Mtop and on JES.
The normalization is set by N (Mtop , JES) that has the
following expression:
4
X
(p3k + p3k+1 · JES
k=0
+p3k+2 · JES 2 ) · (Mtop )k
(27)
The parameters αi (Eq. 26) depend on Mtop and JES as
follows:
p15
i=0
αi =
p3i+13 + p3i+14 · Mtop + p3i+15 · JES i = 1, 2, 3
(28)
In Eqs. 27 and 28 the parameters pi are constants determined from the simultaneous fit of the top templates from all 84 tt̄ samples with the function
Pstop (mtop
Figure 12 shows the function
evt |Mtop , JES).
Pstop (mtop
|M
,
JES)
for
JES = 0 σc and various valtop
evt
ues of Mtop in the case of events with one tagged jet. A
similar parameterization is obtained for events with at
least two tagged jets.
To determine how well the parameterization in Eq. 26
describes the templates, we calculate the χ2 divided by
the number of degrees of freedom, Ndof , as follows
2
χ /Ndof =
P12
m=1
P7
j=1
PN bins
bin=1
Ndof
hbin −fbin
σhbin
2
Mtop:
150 GeV/c2
160 GeV/c2
170 GeV/c2
178 GeV/c2
185 GeV/c2
195 GeV/c2
0.04
0.035
The first set of templates, called the top templates,
is built using a variable (mtop
evt ) determined using the
matrix element technique. We call mtop
evt the event-byevent reconstructed top quark mass, and it represents
the mass value that maximizes the event probability defined in Section IV. We find the value of mtop
evt by evaluating the event probability in the range 125 GeV/c2 →
225 GeV/c2 . When building the templates, we drop the
events for which the event probability is naturally maximized at mass values outside this range. These events
accumulate at the edges of the distribution making difficult the parameterization described below.
For tt̄ events, the function Pstop (mtop
evt |Mtop , JES) used
to describe the shape of these templates is a normalized
product of a Breit-Wigner function and an exponential:
α0 exp −(mtop
evt − α1 )α3
top
top
Ps (mevt |Mtop , JES) =
N (Mtop , JES)
α2 /2π
(26)
× top
(mevt − α1 )2 + α22 /4
N (Mtop , JES) =
Fraction/ 14GeV/c2
17
(29)
where hbin is the bin content of the template histogram
and fbin is the value of the function from Eq. 26 at
the center of the bin. In Eq. 29, the first two sums in
0.03
0.025
0.02
0.015
0.01
0.005
0
130
140
150
160
170
180
190
200
210
220
2
Event Top Mass (GeV/c )
FIG. 12: The function fitting the top templates for tt̄ events
at nominal JES and for various hypotheses of the top quark
mass in the case of events with one tagged jet. A similar parameterization is obtained for events with at least two tagged
jets.
the numerator are over the templates built from simulated tt̄ events for a given Mtop (12 values) and JES
(7 values). The third sum is over all the bins with
more than 5 entries from each template. We obtain
χ2 /Ndof = 1554/1384 = 1.12 for the sample with one secondary vertex tag and χ2 /Ndof = 1469/1140 = 1.29 for
the sample with two secondary vertex tags corresponding to very small χ2 probabilities. From the values of the
quantity χ2 /Ndof , we conclude that the parameterization
of the top templates is not very accurate, and we expect
some bias in the reconstruction of mass and JES. The
procedure for bias removal is described in Section VI C.
The top templates for background events are built using the matrix element in the same way as for tt̄ events.
The shape of the background template is fitted to a normalized Gaussian. Figure 13 shows separately the resulting parameterized curves of background templates for
single and double tagged background events.
The second set of templates, the dijet mass templates,
are formed by considering the invariant mass mW
evt of all
possible pairs of untagged jets in the sample. This variable is correlated to the mass of the W boson, and plays
a central role in the in situ calibration of the jet energy
scale. For tt̄ events the function PsW (mW
evt |Mtop , JES)
used to fit the dijet mass templates is a normalized sum
of two Gaussians and a Gamma function:
"
PsW (mW
evt |Mtop , JES) =
1
N ′ (Mtop , JES)
β6 β7 exp − β7 (mevt W − β8 )
· (mevt W − β8 )β9
×
Γ(1 + β9 )
(mW − β1 )2
β0
+ √ exp − evt 2
2β2
β2 2π
W
(m − β4 )2
β3
(30)
+ √ exp − evt 2
2β5
β5 2π
18
Fraction/ 14GeV/c2
(a)
Fraction/ 14GeV/c2
Background Top Template (1 b-tag)
0.016
0.014
0.012
0.01
0.008
0.024
JES:
-3 σc
-1 σc
1 σc
3 σc
0.022
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.006
0.004
0.004
0.002
0
0.002
130
140
150
160
170
180
190
200
210
20
40
60
80
100
120
140
160
180
200
2
Dijet Mass (GeV/c )
220
2
Event Top Mass (GeV/c )
FIG. 14: The function fitting the dijet mass templates for
tt̄ events with Mtop = 170 GeV/c2 and various values of JES
in the case of events with one tagged jet. A similar parameterization is obtained for events with at least two tagged jets.
(b)
Fraction/ 14GeV/c2
Background Top Template (2 b-tags)
0.013
0.012
0.011
0.01
0.009
0.008
0.007
0.006
0.005
0.004
130
140
150
160
170
180
190
200
210
220
2
Event Top Mass (GeV/c )
FIG. 13: Top templates for (a) single tagged background
events and for (b) double tagged background events.
where the parameters βi depend on Mtop and on JES.
The normalization is set by N ′ (Mtop , JES) that has the
following expression:
N ′ (Mtop , JES) =
1
X
k=0
(q3k + q3k+1 · JES
+q3k+2 · JES 2 ) · (Mtop )k
(31)
The parameters βi depend on Mtop and JES as follows:
βi = q3i+6 + q3i+7 · Mtop + q3i+8 · JES, i = 0, 9
(32)
In Eqs. 31 and 32 the parameters qi are constants determined from the simultaneous fit of the top templates from
all 84 tt̄ samples with the function PsW (mW
evt |Mtop , JES).
Figure 14 shows the function PsW (mW
evt |Mtop , JES) for
Mtop = 170 GeVc2 and various values of JES in the case
of events with one tagged jet. A similar parameterization
is obtained for events with at least two tagged jets.
As in the case of top templates, we calculate (Eq. 29)
the quantity χ2 /Ndof to describe the performance of the
parameterization of the dijet mass templates. We obtain
χ2 /Ndof = 3551/2636 = 1.35 for the sample with one
secondary vertex tag and χ2 /Ndof = 2972/2524 = 1.18
for the sample with at least two secondary vertex tags.
From the values of the quantity χ2 /Ndof we reach the
same conclusion as in the case of the parameterization
of top templates: the parameterization of the dijet mass
templates is not very accurate and some bias is expected
when the top mass and JES are reconstructed.
The dijet mass template for background is built in the
same way as for the tt̄ templates. The background template is fitted to a normalized sum of two Gaussians and
a Gamma function. This combination of functions provided the best fit of the dijet mass shapes. Figure 15
shows separately the resulting parameterized curves of
background templates for single and double tagged background events.
B.
Likelihood definition
The mass of the top quark and the value of JES are
determined by maximizing a likelihood function built using the two sets of templates described in Section VI A.
Assuming that the data sample is the sum of ns tt̄ events
and nb background events, we can calculate the likelihood
function connected to a generic template P f as
Y ns · P f (xevt |Mtop , JES)
s
Lf (Mtop , JES) =
ns + nb
evt=1
!
nb · Pbf (xevt )
+
(33)
ns + nb
tot
Nevt
where index f can either be top when the variable xevt
represents the event-by-event reconstructed top mass, or
W for the invariant mass of pairs of light flavor jets.
The number of tt̄ events, ns , is constrained to the expected number of tt̄ events, nexp
s , via a Gaussian
2
(ns − nexp
s )
(34)
Lns = exp −
2σnexp
s
19
(a)
tags:
Fraction/ 14GeV/c2
Background Dijet Mass Template (1 b-tag)
Ln−tag = Ltop · LW · Lnev · Lns
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
20
40
60
80
100
120
140
160
180
200
2
Dijet Mass (GeV/c )
(b)
Fraction/ 14GeV/c2
Background Dijet Mass Template (2 b-tags)
(36)
As described in Section III, the jet energy scale JES
can be determined from independent detector calibrations. We include this knowledge in the likelihood in the
form of a Gaussian constraint on our variable JES. This
Gaussian has a mean equal to the expectation on JES
from the independent calibration, JESexp = 0 σc , and
a width equal to 1 σc which is the uncertainty on this
expectation.
(JES − JES exp )2
LJES = exp −
(37)
2
The term expressing the constraint on the JES variable is multiplied together with the likelihood function
for each heavy flavor sample to obtain the final likelihood function used to reconstruct the top quark mass
shown in Eq. 38.
0.022
0.02
0.018
0.016
0.014
0.012
0.01
L = L1tag · L2tag · LJES
0.008
0.006
0.004
0.002
0
20
40
60
80
100
120
140
160
180
200
2
Dijet Mass (GeV/c )
FIG. 15: Dijet mass templates for (a) single tagged background events and for (b) double tagged background events.
, the
with mean equal to nexp
and width equal to σnexp
s
s
uncertainty on the expected number of tt̄ events.
The expected numbers of signal events, nexp
s , are 13 for
the single tagged sample and 14 for the double tagged
sample corresponding to a theoretical cross section of
−1
6.7+0.7
.
−0.9 pb [28] and an integrated luminosity of 943 pb
The value of the theoretical cross section assumes a top
are
quark mass of 175 GeV/c2 . The values for σnexp
s
3.7 for the single tagged sample and 3.9 for the double
tagged sample, which take into account both statistical
effects (assuming a Poisson distribution) on nexp
and syss
tematic ones based on the uncertainty on the theoretical
cross section.
The sum of tt̄ and background events, ns + nb , is constrained to the total number of observed events in the
tot
data, Nevt
, via a Poisson probability with a mean equal
tot
to Nevt
Lnev =
tot
tot ns +nb
(Nevt
)
exp(−Nevt
)
(ns + nb )!
(35)
Multiplying the terms expressing the constraints on
the number of events and the likelihood functions for
each template, we obtain separate likelihood functions
for events with one tag and for events with at least two
(38)
Following the maximization of the likelihood function
shown in Eq. 38 we will obtain six numbers: the reconstructed top quark mass Mt , the reconstructed JES variable JESout , and the number of events with different
number of tags for tt̄ , nS1,2 , and for background, nB
1,2 . The
statistical uncertainties on these numbers, δMt , δJESout ,
δnS1,2 , and δnB
1,2 are obtained from the points where the
log-likelihood changes by 0.5.
C.
Calibration of the method
Using samples of simulated tt̄ events and the background sample built based on the model presented in
Section V, we form simulated experiments for a series
of JES and Mtop input values. We then verify that the
reconstructed values of the top quark mass and JES obtained following the maximization of the likelihood function (Section VI B) are in agreement with the input values. The simulated experiments are a mixture of tt̄ events
and background events reflecting the expected sample
composition of the data. In each simulated experiment,
the number of tt̄ events is drawn from a Poisson distribution of mean equal to the expected number of tt̄ events
passing the selection, as determined from simulation (Table VIII). The number of background events is also
drawn from a Poisson distribution with a mean equal
to the difference between the observed number of events
(see Section III, Table I) and the expected number of
tt̄ events.
In order to reduce the statistical uncertainties on potential biases in mass or JES reconstruction, about 10,000
simulated experiments are performed. Due to the finite
size of simulated tt̄ event samples and background sample
the simulated experiments share events between them.
Mtop (GeV/c2 )
150
155
160
165
170
175
178
180
185
190
195
200
Total Observed
Single Tag
18
17
16
16
15
13
14
12
11
9
9
7
48
Double Tag
14
15
14
14
14
14
14
13
11
11
10
8
24
These overlaps result in correlations between the results
of the mass and JES reconstructions from each simulated
experiment. These correlations are taken into account
following the study found in Ref. [36]. The typical value
for the correlation between any two simulated experiments is 6%.
The variables extracted from each simulated experiPE
ment are: the values of mass, MtP E , and JES, JESout
that maximize the likelihood defined in section VI B; the
statistical uncertainties on the above variables, δMtP E
PE
and δJESout
and the pulls for these variables as defined
by
M P E − Mtop
P ullmass = t
δMtP E
PE
JESout − JEStrue
P ullJES =
PE
δJESout
(39)
where JEStrue is the value of JES used in the simulation.
4
3
2
1
0
-1
-2
-3
150
160
170
180
190
200
2
Top Quark Mass Mtop (GeV/c )
FIG. 16: JES versus Top Quark Mass plane. The points represent the reconstructed JES, JESout , and top quark mass Mt
and have attached their corresponding statistical uncertainties, δJESout and δMt . The vertical dashed lines correspond
to the true values of the mass, while the horizontal lines correspond to the true values of JES. For a perfect reconstruction
the points should sit right at the intersection of the dashed
lines.
Eq. 40
Mt = Cm + Sm · (Mtop − 175)
JESout = Cj + Sj · JEStrue
(40)
for Mtop and JEStrue . The parameters Cm , Cj , Sm , and
Sj have the form
Cm
Sm
Cj
Sj
=
=
=
=
a1 + a2 · JEStrue
a3 + a4 · JEStrue
b1 + b2 · Mtop
b3 + b4 · Mtop
(41)
where the parameters {ai } and {bi } from Eq. 41 are
listed in Table IX. They are determined from a linear
fit of the distributions of Cm and Sm versus JEStrue
(Figs. 17 and 18), and of Cj and Sj versus Mtop , respectively (Figs. 19 and 20).
Cm
TABLE VIII: Number of events for samples of simulated
tt̄ events with Mtop ranging between 150 GeV/c2 and
200 GeV/c2 . The numbers correspond to a integrated luminosity of 943 pb−1 , after all selection requirements are made.
The observed number of events is also shown.
JES (σc)
20
176
The distribution of the top quark masses MtP E reconstructed in each simulated experiment is fitted to a Gaussian. The mean of this Gaussian is interpreted as the reconstructed top quark mass of the sample, Mt , while the
width of the Gaussian represents the expected statistical
uncertainty on it, δMt . We apply the same procedure to
determine the reconstructed value of JES, JESout , and
its expected statistical uncertainty, δJESout .
175.5
Figure 16 shows the reconstructed JES and the reconstructed top mass represented by the points, versus the
true JES and true top mass represented by the grid. Ideally the points should match the grid crossings, but there
is a slight bias which has to be removed. The bias is removed in the mass-JES plane by solving the system in
173
175
174.5
174
173.5
-3
-2
-1
0
1
2
3
JES (σc)
FIG. 17: Distribution of parameter Cm (Eq. 41) as a function
of JES.
Sm
21
1.5
TABLE IX: Values of the parameters describing best the linear dependence on the true JES and on the true Mtop , of the
intercept and slope of the Mtop calibration curve and of the
JES calibration curve respectively.
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
-3
-2
-1
0
1
2
3
JES (σc)
Cj
FIG. 18: Distribution of parameter Sm (Eq. 41) as a function
of JES.
0.8
0.4
0.2
(42)
The parameters Xm , Xj , Ym , and Yj from Eq. 42 depend on Mtop and JEStrue as shown in Eq. 43. Solving
Eq. 42 provides the best estimate of the uncertainties on
Mt and on JESout .
0
-0.2
-0.4
-0.6
-0.8
150
160
170
180
190
200
2
Top Quark Mass Mtop (GeV/c )
FIG. 19: Distribution of parameter Cj (Eq. 41) as a function
of Mtop .
Sj
Uncertainty
0.1
0.05
0.008
0.004
0.3
0.002
0.15
0.0008
δMt = Xm · δJEStrue + Ym · δMtcorr
corr
δJESout = Xj · δMtcorr + Yj · δJESout
0.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
Value
175.0
-0.09
0.975
0.016
0.6
-0.003
1.35
-0.0021
corr
and δJESout
.
1
-1
Parameter
a1
a2
a3
a4
b1
b2
b3
b4
150
160
170
180
190
200
2
Top Quark Mass Mtop (GeV/c )
FIG. 20: Distribution of parameter Sj (Eq. 41) as a function
of Mtop .
The uncertainties δMt and δJESout on the reconstructed values Mt and JESout are also affected by the
bias in the reconstruction technique and we need to correct them as well. By differentiating Eq. 40 with respect
to Mtop and JEStrue , we obtain another system of equations to be solved for the corrected uncertainties, δMtcorr
Xm
Ym
Xj
Yj
=
=
=
=
a2 + a4 · (Mtop − 175)
a3 + a4 · JEStrue
b2 + b4 · JEStrue
b3 + b4 · Mtop
(43)
Following the procedure for removing the bias in the
mass reconstruction, the distribution of pull means extracted using simulated experiments (Fig. 21) validates
our bias correction as, on average, the pull mean is estimated to be consistent with zero within the uncertainty.
The width of the pull distribution is used to determine
the corrections on the statistical uncertainties δMtcorr
due to non-Gaussian behavior of the likelihood function
(Eq. 38). Figure 22 shows the mass pull widths versus
top quark mass Mtop . In these plots the JEStrue of the
tt̄ samples is 0 σc . Similar pulls are obtained from tt̄ samples with different values of JEStrue . Based on these figures, it is estimated that the uncertainty on Mt has to
be increased by 11%.
For the reconstruction of JES, Fig. 23 shows the pull
means versus JEStrue , while Fig. 24 shows the pull widths
versus JEStrue . In both plots, Mtop = 170 GeV/c2 . Similar pulls are obtained from tt̄ samples with different values of Mtop . Regarding the bias correction, we reach the
same conclusion as in the case of the mass reconstruction
that, on average, the pull mean is estimated to be consistent with zero within the uncertainties. Based on Fig. 24,
it is estimated that the uncertainty on the JESout has to
be increased by 6%.
In order to further establish the robustness of the technique, the mass and JES are measured in samples for
which the true values are unknown to the authors of this
1
JES Pull Width
Pull Mean
22
Average Pull Mean 0.14±0.22
Central Value
0.8
Uncertainty
0.6
0.4
0.2
Average Pull Width 1.06±0.02
Central Value
Uncertainty
1.15
1.1
1.05
0
-0.2
1
-0.4
-0.6
0.95
-0.8
-1
150
160
170
180
190
200
2
-3
-2
-1
0
1
2
Top Quark Mass Mtop (GeV/c )
Pull Width
FIG. 21: Pull means versus Mtop in the case of the reconstruction of top quark mass in samples with JEStrue = 0 σc .
The continuous line represents the average pull mean and the
dashed lines show the uncertainty on it.
Average Pull Width 1.11±0.03
Central Value
Uncertainty
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
150
160
170
180
190
200
2
Top Quark Mass Mtop (GeV/c )
JES Pull Mean
FIG. 22: Pull widths versus Mtop in the case of the reconstruction of top quark mass in samples with JEStrue = 0 σc .
The continuous line represents the average pull width and the
dashed lines show the uncertainty on it.
1
Average Pull Mean 0.05±0.07
Central Value
Uncertainty
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-3
-2
-1
0
1
2
3
JES (σc)
FIG. 24: Pull widths versus versus JEStrue in the case of the
reconstruction of JES in samples with Mtop = 170 GeV/c2 .
The continuous line represents the average pull width and the
dashed lines show the uncertainty on it.
paper. To validate the mass reconstruction we utilize five
such blind samples: three generated with Herwig and
two with Pythia. The value of JES in these samples
corresponds to 0, the nominal jet energy scale. The reconstructed top quark mass in each of these samples is
the most probable value obtained from 10,000 simulated
experiments. Each simulated experiment is formed combining the tt̄ events in the blind samples and the background events from the background model such that on
average the total number of events is equal to the observed value (see Table VIII). The size of the tt̄ content
is 15 single tagged events and 14 double tagged events.
Following the mass reconstruction technique and
the calibration described in this paper, the differences between the true top quark mass values and
the reconstructed ones are: -0.2, 0.3, 0.6, -0.7, and
1.1 GeV/c2 . The statistical uncertainty on these numbers is 0.8 GeV/c2 . The first two numbers correspond
to the Pythia samples. To validate the JES reconstruction, another five blind samples are used for which the jet
energy scale is modified. The generator used here is Herwig and the value of the top quark mass is 170 GeV/c2 .
The differences between the true JES values and the reconstructed ones are: 0.1, 0.3, 0.0, 0.1, and -0.1 σc . The
statistical uncertainty on these numbers is 0.4 σc .
In conclusion, both the mass and JES reconstructed
values are compatible with true ones within the statistical
uncertainties. This additional check gave us confidence
that the method described here can be reliably applied
on the data to reconstruct JES and the top quark mass.
3
JES (σc)
D.
FIG. 23: Pull means versus JEStrue in the case of the reconstruction of JES in samples with Mtop = 170 GeV/c2 .
The continuous line represents the average pull mean and the
dashed lines show the uncertainty on it.
Expected statistical uncertainty
In Fig. 25 we show
quark mass, δMtcorr ,
JEStrue = 0 σc . Since
depends on the Mtop ,
the expected uncertainty on top
versus Mtop , for samples with
the expected number of tt̄ events
the uncertainty δMtcorr depends
removing the parameterization as a function of JES and
by maximizing a likelihood built only with the top templates corresponding to JES = 0 σc . We reconstruct the
top quark mass for two samples with Mtop = 170 GeV/c2 ,
but with different values for JEStrue : +1 σc , and -1 σc ,
respectively. Taking half of the difference between the
two reconstructed Mt determines the systematic uncertainty due to jet energy scale as 2.2 GeV/c2 , which is 10%
more than in the case of using the in situ calibration and
the JES parameterization.
2
Uncertainty on Mt (GeV/c )
23
4.5
4
3.5
3
2.5
160
165
170
175
180
185
190
2
Top Quark Mass Mtop (GeV/c )
VII.
Uncertainty on JES (σc)
FIG. 25: Expected uncertainty on top quark mass, δMtcorr ,
versus Mtop , for samples with JEStrue = 0 σc . This uncertainty includes the uncertainty due to statistical effects and
the systematic uncertainty due to jet energy scale.
1
0.98
0.96
0.94
0.92
0.9
0.88
0.86
0.84
0.82
-3
-2
-1
0
1
2
3
JES (σc)
FIG. 26: Expected uncertainty on JES, δJESout , versus
JEStrue , for samples with Mtop = 170 GeV/c2 .
on it too. Figure 26 shows the expected uncertainty on
corr
JES, δJESout
, versus JEStrue , for samples with Mtop
= 170 GeV/c2 . The uncertainties in Fig. 25 and 26 are
corrected for bias, but not for pull widths (non-Gaussian
effects).
The expected uncertainties shown in Fig. 25 contain
both the statistical uncertainty on the top quark mass
and the uncertainty due to jet energy scale. In order to
disentangle the statistical uncertainty on Mt from the one
due to jet energy scale, we reconstruct the top quark mass
by maximizing the likelihood for a fixed value of JES. Following this reconstruction for Mtop = 170 GeV/c2 and
JEStrue = 0 σc , the uncertainty on the top quark mass
is 2.5 GeV/c2 . In comparison, when JES is not fixed
the expected uncertainty (Fig. 25) on Mt is 3.2 GeV/c2 .
Subtracting these two numbers in quadrature we estimate that the systematic uncertainty on Mt due to jet
energy scale is 2.0 GeV/c2 .
We can determine the systematic uncertainty on Mt
due to the jet energy scale in the absence of the in situ
calibration (provided by the dijet mass templates), by
RESULTS
Applying the event selection described in Section III
to the multijet data corresponding to an integrated luminosity of 943 pb−1 , we observe 48 events with one secondary vertex tag and 24 events with at least two secondary vertex tags. Performing the likelihood maximization and applying the corrections described in Section VI
for this sample, we measure a top quark mass of 171.1 ±
3.7 GeV/c2 and a value for JES of 0.5 ± 0.9 σc .
Figure 27 shows the distributions of reconstructed top
quark masses for data (dots) and for the combination
(light) of signal and background templates that best fit
the data. The background (dark) contribution is shown
normalized to the data as determined by the fractions
obtained from the likelihood fit. There are two sets of
distributions corresponding to the sample with only one
secondary vertex tag (Fig. 27(a)) and to the sample with
at least two secondary vertex tags (Fig. 27(b)).
The minimized negative log-likelihood is shown in
Fig. 28 as a function of the top mass and JES after correcting for bias (Eqs. 40 and 42) and for non-Gaussian
effects (Section VI C). The central point corresponds to
the minimum of the negative log-likelihood, while the
contours are given at a number of values of ∆lnL, the
change in negative log-likelihood from its minimum.
Table X lists the number of events for tt̄ and for background for the one- and two-secondary vertex tags cases,
as measured following the minimization of the two dimensional likelihood of Eqn. 38 on the data.
TABLE X: Measured sample composition of the multijet data
sample for a luminosity of 943 pb−1 , passing the event selection. The second column (1 tag) gives the number of events
with only one secondary vertex tag, while the third column
(≥2 tags) is for the events with at least two secondary vertex
tags.
Number of Events
Signal (tt̄ )
Background
Total Observed
1 tag
13.2 ± 3.7
34.6 ± 7.2
48
≥2 tags
14.1 ± 3.4
9.2 ± 4.3
24
Using a tt̄ Monte Carlo sample with a top quark mass
equal to 170 GeV/c2 and the number of signal and back-
24
the measured value of 3.7 GeV/c2 . This can be seen in
Fig. 29, where the histogram shows the results of the simulated experiments and the vertical line represents the
measured uncertainty. In conclusion, the measured combined statistical and JES uncertainties on the top mass
agree with the expectation.
14
12
10
8
6
4
2
0
140
160
180
200
220
2
Mtop of Event (GeV/c )
Events/ 10GeV/c2
(b)
10
Entries/ 0.3 GeV/c2
Events/ 10GeV/c2
(a)
16
1800
1600
Expected Uncertainty
1400
Measured Uncertainty
1200
1000
800
600
400
8
200
0
0
6
2
4
6
8
10
12
14
2
Statistical Uncertainty on Mtop (GeV/c )
4
2
0
140
160
180
200
220
2
Mtop of Event (GeV/c )
JES (σc)
FIG. 27: Reconstructed top mass for data (points), best signal+background fit (light) and background shape from the
best fit (dark): for (a) samplee with only one secondary vertex tag, and (b) the sample with at least two secondary vertex
tags.
3
2
∆ ln L=4.5
∆ ln L=2.0
1
∆ ln L=0.5
FIG. 29: Distribution of expected statistical uncertainty on
Mt (histogram) and the measured uncertainty (vertical line).
In about 41% of simulated experiments a statistical uncertainty on the top quark mass smaller than in the experiment
is found.
In order to obtain the contribution of the uncertainty
in jet energy scale to the uncertainty on the top quark
mass, the minimization of the 2D likelihood is modified
such that the JES parameter is fixed to 0.5 σc (the value
of JES from the likelihood minimization). Following this
procedure the uncertainty on the top mass is 2.8 GeV/c2 .
Subtracting in quadrature this value from the uncertainty
obtained when the JES was not fixed (3.7 GeV/c2 ), we
estimate the systematic uncertainty contributed by JES
as 2.4 GeV/c2 .
0
-1
VIII.
-2
-3
160
165
170
175
180
185
2
Top Mass (GeV/c )
FIG. 28: Contours of the likelihood in the Mtop and JES
plane at a number of values of ∆lnL, the change in negative
log-likelihood from its maximum.
ground events from Table X, we perform simulated experiments and determine the distribution of expected uncertainty on the top quark mass due to statistical effects
and JES. About 41% of the simulated experiments have a
combined uncertainty on the top quark mass lower than
SYSTEMATIC UNCERTAINTIES
We model tt̄ events using simulated events, which do
not always accurately describe all effects we expect to see
in the data. The major sources of uncertainties appear
from our understanding of jet fragmentation, our modeling of the radiation from the initial or final partons,
and our understanding of the proton and antiproton internal structure. Apart from these uncertainties, which
are present in most top quark measurements, we also address other issues specific to the present method such as
the shape of the background top templates following the
correction for tt̄ content, and the uncertainty in the two
dimensional correction of the reconstructed top mass and
JES.
25
A.
Systematic uncertainties related to jet energy
scale
1.
b-jet energy scale
We study the effect of the uncertainty on the modeling
of b quarks due to the uncertainty in the semi-leptonic
branching ratio, the modeling of the heavy flavor fragmentation, and due to the color connection effects.
To determine this we reconstruct the top mass in a
simulated tt̄ sample (Mtop = 175 GeV/c2 ) where we select
b-jets by matching the
p b quarks to a jet. The matching
procedure requires (∆η)2 + (∆φ)2 < 0.4 between the
quark and the jet. We modify the energy of the b-jets
by 0.6% corresponding to the uncertainty on the b-jet
energy due to the effects listed above [37]. The resulting
systematic uncertainty on the top quark mass due to the
uncertainty on the b-jet energy scale is 0.4 GeV/c2 .
2.
Systematic uncertainties due to background
1.
Source of Systematic
δMt (GeV/c2 )
Response Relative to Central Calorimeter
0.2
Multiple Interactions
0.1
Modeling of Hadron Jets
0.5
Modeling of the Underlying Event
0.0
Modeling of Parton Showers
0.5
Energy Leakage
0.1
Total Residual JES Uncertainty
0.7
change in the value of the reconstructed top quark mass
by 0.9 GeV/c2 which is added as a systematic uncertainty.
Residual jet energy scale
From the two dimensional fit for mass and JES, we extract an uncertainty on the top quark mass that includes
a statistical component as well as a systematic uncertainty due to the uncertainty on the jet energy scale.
This systematic uncertainty is a global estimate of the
uncertainty due to jet energy scale. Additional detailed
effects arise from the limited understanding of the individual contributions to JES (see Section VI).
For this we have to study the effect on the top mass
reconstruction from each of these sources: angular dependence of the calorimeter response, contributions by
multiple interactions in the same event bunch, modeling
of hadron jets, modeling of the underlying event, modeling of parton showers and energy leakage. A simulated
tt̄ sample (Mtop = 175 GeV/c2 ) is used where the energies of the jets have been shifted up or down by the
uncertainty at each level separately. We reconstruct the
top quark mass for each case, without applying any constraint on the value of JES. Table XI shows the average shift on the top mass at each level, and the sum in
quadrature of these effects. We conclude from this study
that the uncertainty on the top quark mass contributed
by these corrections to the jet energy is 0.7 GeV/c2 .
B.
TABLE XI: Residual jet energy scale uncertainty on the top
mass. The sum in quadrature of all the effects represents the
total residual systematic uncertainty due to jet energy scale.
Background modeling
Based on the background model (Section V), we assume Mtop = 170 GeV/c2 to correct for the presence
of tt̄ events in the background distributions. To estimate the uncertainty associated with making this assumption, we modify our background model considering
a 10 GeV/c2 variation on Mtop used in the default background correction procedure. This variation results in a
2.
Background statistics
Another effect we address here is that of the limited
statistics (≈ 2600 events, see Section V) of the data sample used to model the background. To estimate this effect
we vary the parameters describing the background templates within their uncertainties. Using the procedure
described below, we find that the effect on the reconstructed top quark mass due to variation on the background dijet mass templates is negligible. This is not the
case of the background top templates.
For simplicity, we label the parameters of this template
as Constant, Mean and Sigma, representing the constant,
the mean and the width of the Gaussian function describing the background top template. In order to find the
uncertainties on these parameters, we vary the content
of the top template histograms for background assuming
that each bin fluctuates according to a Poisson probability. This variation is done 10,000 times, and each time
we extract and form distributions with the values of the
three parameters, Constant, Mean and Sigma after applying the correction due to the residual tt̄ content in the
sample. We use the spread of these distributions as the
uncertainties on the parameters of the top templates for
background.
Table XII shows the values of these uncertainties separately for the sample with only one secondary vertex tag
(1tag) and for the sample with at least two secondary
vertex tags (2tags). Varying the parameters of the background top templates within these uncertainties results in
a shift in the reconstructed top quark mass of 0.4 GeV/c2
and we add this as a new systematic uncertainty.
26
TABLE XII: Parameters of the top templates for background
events. These templates have been described in Section VI A.
The second column is for the single tagged sample (1tag),
while the third column is for the double tagged sample (2
tags).
Parameter
Constant ((GeV/c2 )−1 )
Mean (GeV/c2 )
Sigma ((GeV/c2 )2 )
C.
1 tag
0.015±0.001
159±3
1790±272
2 tags
0.013±0.001
163±3
3280±712
Initial and final state radiation
The top quark mass measurement is affected by how
we model the initial and final state gluon radiation. This
radiation affects the jet multiplicity in the event as well
as the energy of the jets, which in turn affect the top
quark mass reconstruction.
The amount of radiation from the initial partons is
controlled in our simulated tt̄ samples by the DGLAP
evolution equation [38] [39]. The parameters of these
equations are ΛQCD and K (the scale of the transverse
momentum for showering). In the case of the initial state
radiation, these parameters are tuned in the simulation
to reflect the amount of radiation observed in Drell-Yan
events [37]. The amount of radiation, proportional to the
average transverse momentum of the leptons, is found to
depend smoothly on the invariant mass of the leptons,
over a range of energies extending up to the range of
tt̄ events. Two sets of values for the parameters ΛQCD
and K are determined to cover the variation of this dependence within one standard deviation (σISR ).
We generate two samples of tt̄ events (Mtop =
178 GeV/c2 ) where the parameters ΛQCD and K correspond to +σISR (increase the amount of radiation), and
−σISR (decrease the amount of radiation), respectively.
Using the default set of values, the reconstructed top
quark mass is 178.6 GeV/c2 . For the sample with +σISR
the reconstructed top quark mass is 178.9 GeV/c2 , and
for the one with −σISR the reconstructed top quark mass
is 178.6 GeV/c2 . Taking the maximum change in top
mass, we quote 0.3 GeV/c2 as the uncertainty due to
initial state radiation modeling.
Using the same variation of the parameters ΛQCD and
K to describe the variation of the final state radiation,
we reconstruct the top quark mass to be 177.7 GeV/c2
in a sample with increased radiation and 177.4 GeV/c2
when we decrease the amount of radiation. Taking into
account the value of the reconstructed top quark mass
in the default case, the maximum change in the reconstructed top quark mass is 1.2 GeV/c2 representing the
systematic uncertainty on the modeling of the final state
radiation.
D.
Proton and antiproton PDFs
In our default simulation, the internal structures of
the proton and antiproton are given by the cteq5l set
of functions, and for a tt̄ sample with Mtop = 178 GeV/c2
the reconstructed top quark mass is 178.6 GeV/c2 . For
the same Mtop value, using a different set of functions
(cteq6m) results in a reconstructed top quark mass of
178.7 GeV/c2 . Within the cteq6m set, there are 20 independent parameters whose uncertainties are representative of the uncertainty on the modeling of such structure
functions [40]. Adding in quadrature all the 20 offsets
observed in top quark mass reconstruction due to these
variations, we get 0.4 GeV/c2 .
Also, it is known that the value of ΛQCD has a direct
effect on the shape of the structure functions. In order
to estimate this effect, we chose yet another set of PDFs
given by MRST, and reconstructed the top mass for
ΛQCD = 228 GeV to get a top mass of 177.4 GeV/c2 , and
for ΛQCD = 300 GeV to get a top mass of 177.7 GeV/c2 .
Therefore the systematic uncertainty due to the value of
ΛQCD is 0.3 GeV/c2 .
Adding the two contributions in quadrature, we
quote that the total systematic uncertainty due to the
choice of structure functions of proton and antiproton is
0.5 GeV/c2 .
E.
Other systematic uncertainties
The default Monte Carlo generator used to determine
our templates is Herwig, which is known to differ from
the Pythia generator. For simulated tt̄ samples with
Mtop = 178 GeV/c2 , we reconstruct the top quark mass
as 177.6 GeV/c2 using Herwig as the generator, and
178.6 GeV/c2 using Pythia. We assign a systematic uncertainty due to the choice of the Monte Carlo generator
of 1.0 GeV/c2 representing the difference between the reconstructed top quark masses in Herwig and Pythia.
In addition, we have varied the parameters of Eq. 40
within their uncertainties as listed in Table IX, and obtained new values of the top quark mass. The changes
from the default value are within 0.2 GeV/c2 .
F.
Summary of the systematic uncertainties
The total systematic uncertainty on the top mass combining all the effects listed above is 2.1 GeV/c2 . Table XIII summarizes all sources of systematic uncertainties with their individual contribution as well as the combined effect.
IX.
CONCLUSION
We measure the mass of the top quark to be
171.1 GeV/c2 with a total uncertainty of 4.3 GeV/c2 .
27
TABLE XIII: Summary of the systematic sources of uncertainty on the top mass. The sum in quadrature of all the
effects represents the total systematic uncertainty.
Source
b-jet JES
Residual JES
Background Modeling
Background Statistics
Initial State Radiation
Final State Radiation
pp̄ PDF Choice
Pythia vs. Herwig
Method Calibration
Sample Composition
Uncertainty (GeV/c2 )
0.4
0.7
0.9
0.4
0.3
1.2
0.5
1.0
0.2
0.1
Total
2.1
quark mass measurement.
As the luminosity collected with the CDF II detector
increases to an expected 7 fb−1 for Run II, the statistical
uncertainty on the top quark mass will improve and additional top quark mass results from CDF are expected in
the near future. A more careful estimation of the sources
of systematic uncertainties on the top quark mass as well
TABLE XIV: Most precise results from each tt̄ decay channel
from the Tevatron by March 2007. The integrated luminosity
used in these analyses is about 1 fb−1 .
Channel
Lepton+Jets [3]
Dilepton [4]
All-hadronic (this analysis)
All-hadronic (previous result) [2]
Result
170.9±2.5 GeV/c2
164.5±6.5 GeV/c2
171.1±4.3 GeV/c2
174.0±5.3 GeV/c2
This measurement, the most precise to-date in the allhadronic channel, is performed using 943 pb−1 of integrated luminosity collected with the CDF II detector.
This is the first simultaneous measurement of the top
quark mass and of the jet energy scale in the tt̄ allhadronic channel. It is also the first mass measurement in
this channel that involved the use of the tt̄ matrix element
in the event selection as well as in the mass measurement
itself.
The previous best mass measurement published in this
channel, for an integrated luminosity of 1 fb−1 , has an
equivalent total uncertainty of 5.3 GeV/c2 [2] which is
23% more than in this measurement. The main source
for the observed improvement is the reduction of the
uncertainty on the top quark mass due to jet energy
scale (JES). In the present analysis, this uncertainty
is 2.5 GeV/c2 (including the residual JES uncertainty
of 0.7 GeV/c2 ), which is about twice smaller than the
corresponding uncertainty of 4.5 GeV/c2 determined in
Ref. [2].
The top quark mass measured in this analysis is
consistent with the most precise top quark mass values measured at the Tevatron and at CDF in the lepton+jets [3] and the dilepton [4] channels. This consistency among the decay channels restricts the possibility for new physics to prefer the tt̄ all-hadronic decay
channel over the other decay channels. Table XIV summarizes the most precise top quark mass measurements
made at the Tevatron using an integrated luminosity of
about 1 fb−1 . From this table it can be seen that the allhadronic channel provides the second most precise top
as a more efficient tt̄ event selection can help to further
reduce the total uncertainty in this analysis. We expect
that future mass measurements performed in this channel
using an increased data sample size will improve the total
uncertainty on the top quark mass which will contribute
to our understanding of the electroweak interaction as
well as to the search for new physics.
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Acknowledgments
We thank the Fermilab staff and the technical staffs
of the participating institutions for their vital contributions. This work was supported by the U.S. Department
of Energy and National Science Foundation; the Italian
Istituto Nazionale di Fisica Nucleare; the Ministry of
Education, Culture, Sports, Science and Technology of
Japan; the Natural Sciences and Engineering Research
Council of Canada; the National Science Council of the
Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium
für Bildung und Forschung, Germany; the Korean Science and Engineering Foundation and the Korean Research Foundation; the Science and Technology Facilities
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the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovación, and Programa ConsoliderIngenio 2010, Spain; the Slovak R&D Agency; and the
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